What could be some potentially useful mathematical databases? This is a soft question but it's not meant as a big-list question. I have recently been asked whether I want to provide feedback at the pre-beta stage on a forthcoming website that will provide a platform for data sharing, and rather than giving just my personal opinion I'd rather consult other mathematicians first. I was going to write a blog post but then I thought that Mathoverflow was a more suitable place since I have a question and I'm looking for answers of a certain type rather than general comments. The website seems to be aimed mostly at scientists who want to share raw data, so at first I thought it probably wouldn't be much use to mathematicians since our data is (or are if you prefer) mostly highly interlinked -- the connections are often more interesting than what they connect. 
But on further reflection, it seems to me that a good data sharing site could be a valuable resource, even if it doesn't do absolutely everything any mathematician would ever want. For instance, Sloane's database is fantastically useful. A rather different sort of database that is also useful is Scott Aaronson's Complexity Zoo. So useful databases exist already. Is this an aspect of mathematical life that could be greatly expanded given the right platform? And if so, what should the platform be like?
I don't know anything about the design of the site, but if I'm going to comment intelligently on what features it would need to have to be useful to mathematicians, I'd like to be armed with some examples of the kind of data sharing we might actually go in for. Here are a few ideas off the top of my head. 


*

*Diophantine equations: one could have a list of what is known about various different ones.

*Mathematical problems: listed in some nice categorized way, each problem accompanied by a description, complete with reading list, of what you really ought to know before thinking about the problem. (As an example, if you are thinking about the P versus NP problem, then you really ought to know about the Razborov/Rudich natural proofs paper.)

*Key examples in various different areas and subareas of mathematics.

*Sometimes you have a whole lot of related mathematical properties with a complicated pattern of implications between them. Under such circumstances, it could be nice to have this information presented in a nice graphical way (something I think this site may be able to do well -- they seem to be keen on visualization) with links to proofs of the implications or counterexamples that demonstrate when the implications do not hold. (The example I'm thinking of while writing this is different forms of the approximation property for Banach spaces, but there are presumably several others.)

*List of special functions and the facts about each one that are the main facts one uses to prove things about them.

*List of integrals that can be evaluated, with descriptions of how they can be evaluated.

*List of important irrational numbers with their decimal expansions to vast numbers of places. (I'm not sure why this would be useful but it might be amusing.)
These are supposed to be examples where people could usefully pool the background knowledge that they pick up while doing research. I'm not particularly pleased with them: they should be thought of as a challenge to come up with better ones, which almost certainly exist. If you've ever thought, "Wouldn't it be nice if there's somewhere where I could look up X," then X would make a great answer. I think the most interesting answers would be research-level answers (unlike some of the suggestions above).
If there were a site with a lot of databases, it would make a great place to browse: it would be much easier to find useful data there than if it was scattered all round the internet.
One constraint on answers: there should be something about a suggested database that makes it unsuitable for Wikipedia, since otherwise putting it on Wikipedia would appear to be more sensible.
 A: For numerical analysts and scientific computing folks, a database of 'standard problems' with given geometry, parameters, tolerances and input/output specifications,  and a mechanism for storing and comparing (curated) computational attacks on these. For example, the problem of lid-driven cavity flow is considered a major test for computational fluid dynamics, and it would be great to have an agreed set of 3 or 4 sub-problems (laminar flows, angled walls, incompressible flows, nearly incompressible flow) on which the performance of algorithms could be compared. 
Comparisons of the performance of algorithms on a given problem according to criteria such as accuracy, storage needed, and efficiency would be useful. Code in a given language would be useful, but one probably cannot insist on this.   
There's a dearth of such 'standard problems', and thus algorithms purportedly approximating solutions for the same problem are rarely compared. 
NIST has an example of such standard problems for  models of micromagnetics.
http://www.ctcms.nist.gov/~rdm/mumag.org.html
A: Having a list of named theorems along with their statements (and references) would be great.
Here's by a "named theorem", I mean a theorem that is routinely quoted by experts in the field, without giving any reference or further explanation, such as "Noether's theorem".
A: It appears that none of us suggested exactly this in response to Timothy's call, but 
a group of authors from Germany, Denmark, and Australia
have committed to make it happen:
"The Open Graph Archive: A Community-Driven Effort."
From their conclusion:

We advocated the need for an open, worldwide graphbase to collect and distribute graphs and programs for their generation, analysis, manipulation, and drawing.

And here is their Abstract:

In order to evaluate, compare, and tune graph algorithms, experiments on well designed benchmark sets have to be performed. Together with the goal of reproducibility of experimental results, this creates a demand for a public archive to gather and store graph instances. Such an archive would ideally allow annotation of instances or sets of graphs with additional information like graph properties and references to the respective experiments and results. Here we examine the requirements, and introduce a new community project with the aim of producing an easily accessible library of graphs. Through successful community involvement, it is expected that the archive will contain a representative selection of both real-world and generated graph instances, covering significant application areas as well as interesting classes of graphs.

Visit http://graphdrawing.org/grapharchive/ to see their nascent effort.
A: Several times this week I wished I could find a list of adjunctions and the explicit monad and comonad thereby generated.  May as well throw in the algebras thus generated, as well as the Kleisli and Eilenberg-Moore categories.  The usefulness would come from having all the definitions properly unwound, so that one would recognize the 'familiar' objects immediately.
A: A list of theorems (and the formalization) that have been formalized in some proof assistant e.g. coq. Maybe not so much for now but for the future. s.t. work is not unnecessarily done to often.
A: I would love to see a database of theorems published online with links to the theorems they depend on. For example, if I looked up the Hahn-Banach theorem it'd point me to the Riesz extension theorem and any other major theorems required in its proof. If there are multiple published proofs, then more than one could be listed, each with its own set of links to previous theorems.
An application might be something like this: if I wanted to learn something like the Atiyah-Singer index theorem, it'd look it up, and the database could generate a complete dependency graph (or multiple such graphs) of major theorems I need to learn before I can get there. The database could even be personalised so it knows that it knows what theorems I'm familiar with and can use that to find an optimal path for me to learn something new,
In principle, any published book or paper would be representable as a subgraph of the full database graph. So it'd be nice to be able to look up a paper in this database and see that, in essence, this paper proves X, Y and Z using results A, B and C from papers P, Q and R.
Ultimately such a thing could morph into something where the proofs and dependencies are represented formally, but we don't have the technology for that yet.
A: My feeling about this is closest to the answer by Timothy Chow. First somewhere I can look up a definition or a named theorem would be better off somewhere else. There are existing databases that have proved themselves useful. The OEIS is far more useful than I would ever have expected. John Cremona's database of elliptic curves is the result of a lifetime's hard work although I have not used it myself. Again it is not my field but my understanding is that GAP includes many databases that have been built up over years. There is also Thistlewaite's KnotScape and Bar Natan's KnotAtlas which are databases of knot tables and knot invariants. There is also Nauty which produces great lists of graphs and related structures.
It seems to me that all of these become far more useful if the database is part of a computer algebra package. Furthermore that it is not just the data but the calculations that can be done with it that make it useful to mathematicians.
My feeling is that 5. 6. 7. are covered already in computer algebra packages. 
Edit: Of course I left out the two databases that I use without thought. MathSciNet and the arXiv.
A: Sorry I couldn't resist mentioning the answers to your own question as a viable database:


*

*Examples of common false beliefs in mathematics
(Of course, before it becomes a typical database, it will require some organizing, categorizing, etc.)
A: I agree with André Henriques regarding algebraic topology. In addition to cohomology of $K(\mathbb{Z},n)$ data, I'd also like to see pages of spectral sequences computing various other things of interest, e.g. homotopy groups of spheres, complex cobordism, modules over real $K$-theory, etc. This subject has a massive dearth of examples, especially of examples using spectral sequences. One place where some computations exist is Bob Bruner's webpage, and I like the way he displays some of those spectral sequences as JPEGs. I envision something like that going into this database, but with more computations because more mathematicians can contribute.
In a related vein, I would LOVE to have a repository of applets and other computer code mathematicians have written to help them do computation. For instance, Aaron Mazel-Gee has an Adem relation calculator which would have helped me a ton back when I was starting to learn this material. I've found various other applets (e.g. from Christian Nassau's webpage linked from Bob Bruner's page) but it's not easy. If we had a database it seems it would be a natural place to put up code we've written along with documentation to help others with similar computations. Obviously algebraists have SAGE and GAP but there seems to be nothing like that for algebraic topology. If there is such a place please let me know, especially if it helps with the bookkeeping in spectral sequence computations!
A: KnotInfo.  The three-manifold and knot censuses included with SnapPy.  Also relevant is CompuTop, a list of "topological" software. 
A: It would be really useful to have a database containing various computations of (co)homology/homotopy groups of various spaces that arise in algebraic topology... Note:
There is so much known out there that one would have to first think really hard about how to organize it all.
Here's an example:
I could imagine that, for certain users, listing the first 30 integral cohomology groups of the spaces $K(\mathbb Z,1)$, $K(\mathbb Z,2)$, $K(\mathbb Z,3)$, and $K(\mathbb Z,4)$
could be more useful¹ than listing all the cohomology groups of all the $K(\mathbb Z,n)$'s. The reason is that, in order to do the latter, the information has to be packaged in a certain way that might be hard to understand: the user would need to unpack that information before she can access it.

¹ Of course, it's even better to have both pieces of information available.
A: I like that this is community wiki, so I can throw out a crazy idea and not feel bad if people vote it down. I've come across a number of questions on MO of the flavor "please give me a textbook recommendation for subject X." What about having a page in the database for mathematicians to rank textbooks in various fields, e.g. by voting for them. 
The obvious problem I foresee with this is that different textbooks are good for different levels of learning, but I suppose this page could be geared towards undergrads or early grad students who don't yet know which books will serve them best. By the time you are an expert in a field you don't need textbook recommendations, so this aspect of the database would need to be purely pedagogical and geared towards people starting in a subject. 
As it stands, I need to search several websites (e.g. Google books, Amazon, Goodreads, etc) for recommendations and these can come from literally anyone. With this database I assume we'd be restricting who can post to the database so that only professional mathematicians could. So I would know that I could trust the recommendations as coming from a mature and knowledgeable source rather than an undergrad who just got a bad grade and blames the textbook.
A: In the flavor of "Wouldn't it be nice if there's somewhere where I could look up X," I think the database could probably use a list of online lectures which are freely available, e.g. some entries given by this MO question.
In the same vein, I could also imagine listing lecture notes people have prepared for courses, but this would be harder since there are so many places online where professors have posted their lecture notes and it can be difficult to say when one set of lecture notes is "better" than another. Still, having online lectures as well as resources for teaching (e.g. calculus applets) would fit the bill of things I like to look up but often have to search all over the web for.
A: Falling under "Wouldn't it be nice if there's somewhere where I could look up X":
Typos, mistakes, and expanded calculation for published papers. Of course, this is more or less wishful thinking since I have no idea how to deal with potential correctness issues, etc...
A: There are several natural examples from set theory. 


*

*Here is a database on consequences of and equivalent formulations of the axiom of choice, which is searchable by keyword and axiom form, and which contains hundreds of formulations of choice-like axioms, while providing the implication relations between them and the known models that separate them. I have had occasion to use this database in my research, and the data is useful. I can easily imagine, however, a greatly improved interface or more visual manner of presenting the data, and perhaps this is the kind of situation you seek.

*It would be useful to have a nice visual database of the connections and possibilities between all the various cardinal characteristics of the continuum. Perhaps such a thing exists already...

*More generally, I can imagine some kind of database keeping track of the various independence results over ZFC (or other theories), especially when combinations of statements are considered, and the models that achieve them.

*I can imagine a database of the logical relations between various large cardinal notions and their strengths. The mere fact that the large cardinal chart in Kanamori's book The Higher Infinite is so often consulted proves that a much larger and more comprehensive version of that information would be useful. 
A: I would like a mathematical search-thesaurus;  this would be a list of descriptions people thought of to use as search terms (and that I might think of using) and for each description, a list of phrases that appeared in documents which contained stuff relevant to the descriptions.  A recent example might be: "gently falling curve" yields "roller coaster physics".  A prototype of a thesaurus could be built out of the MathOverflow database.
Gerhard "Searching For The Right Words" Paseman, 2011.06.21
A: The one thing that can really be a problem, particularly when you are not that familiar with some area, is when you become interested in objects of some kind with a particular property and you do not know that these are usually being referred to as, say, Garfield-Dilbert-FooBars. More generally speaking, sometimes you do not know the name of a mathematical definition and that name alone would help immensely. I for one would like to have a hierarchical database where you can find such names. For instance, you search for "surface", "complete" and "regular" and (among maybe other things), it tells you what a K3 Surface is. There should be plenty more examples like this.
A: "6. List of integrals that can be evaluated, with descriptions of how they can be evaluated."
I think it would also be useful to have a list of integrals that can't be evaluated, functions whose antiderivatives can't be expressed in closed form in terms of powers, logs, exponentials, trig functions and the like.
A: Lie group theory and their representations certainly deserve a database. (See e.g. Atlas project and Olver's books that are the only reference for classification of smooth group actions on $\mathbb{R}^n$ for small $n$ that I am aware of.)
Also there is a project to create an internet database of solutions to Einstein equations and I guess that other important PDE's could use a database too.
A: I'd love to have a database of various combinatorial structures. 
For example, a while ago Greg Kuperberg found for me the small ones here, and I joyfully have in my possession 9 DVDs (around 40 gigabytes, in a very efficient special purpose format) containing the 11,084,874,829 triple systems on 19 points as found by [P. Kaski and P. R. J. Östergård, The Steiner triple system of order 19, Math. Comp. 73 (2004), 2075–2092.] 
A: I would like such a database to include a list of conferences by field of study and location of the conference. As it stands several people have webpages where they post about conferences and there's also the AMS page but some conferences still slip through and I don't hear about them till they're over. One would assume that a giant database which all professional mathematicians could contribute to would not have this problem. Conference organizers could post to this database and in this way get the word out to a much larger audience. It's also not at all hard (using say SQL) to make your database self-cleaning, i.e. remove conferences that have already happened.
A: Here is a model to emulate: The Complexity Zoo, and the associated Complexity Garden,
already mentioned by Timothy.
If you ever encounter an unknown (to-you) complexity class acronym (PPP, LOCCFL, QIP, ...), then this is the source to consult.  Does this fail the Wikipedia test?  You can look up each acronym individually, but
to have the entire Zoo (I count[grep] 569 entries) in front of you as you try to identify features of the animals is extremely
useful. And in fact most Wikipedia entries for the individual classes link to The Complexity Zoo
for further information.
A: There are two databases I have wished for during my studies. 
One of notation for various mathematical concepts, covering cultural differences. For example the different ways to denote the open-closed interval, or that different symbols are used for strict subset. LaTeX symbols or macros when they exist, would also be useful. This would have greatly helped me when reading things for the first time and later with writing. Such material is definitely out of scope for Wikipedia. 
Another useful database would be of logics and logical theories. These could range from propositional logics to higher order and infinitary logics. The theories should include various fragments of arithmetic, algebraic theories, theories of strings, etc. For each, I would be interested in what is known about decidability (and computational complexity), completeness, interpolation, etc. Currently, I use the Stanford Encyclopaedia of Philosophy and Wikipedia but what I want may often not be there or not succinctly presented for reference purposes.
A: I feel this proposed database would need a link to the Online Encyclopedia of Integer Sequences
This seems to fit the bill as "the kind of data sharing we would go for," although it would be rather silly to duplicate the OEIS rather than simply linking to it.
A: It seems to me that it would be a good start to list existing resources that are already known to be useful.  These should be a good indication of what other resources would be useful, if not examples of resources that would benefit from being located in a single centralized place instead of scattered across the web.  Here are three categories of resources that come to mind.


*

*Tables of data for small examples of various important families of mathematical objects.  In addition to those that have already been listed, I'll mention Cremona's elliptic curve data and a list of data about small matroids on David Haws's site.

*Code repositories for implementations of important algorithms that aren't readily available in a standard mathematical computation package.  Currently these seem to be scattered across people's personal homepages, and run on a variety of computing environments, although Sage is making a creditable attempt to unify everything.

*Dynamic surveys, such as those maintained by the Electronic Journal of Combinatorics.  These are typically maintained by a single person, although some of them might benefit from wikification.  The NP Optimization Compendium might be an example of this; the subject is so vast that it is hard for just one or two people to keep it up to date.
A: List of theorems, with tagged input/output. The way I see it,
is that each lemma/theorem/proposition has some input (objects they state something about),
and output (the actual result).
For example, a simple theorem that states
"A polynomial in one variable with non-negative coefficients do not have positive roots."
could be tagged as
Input: Polynomial P with properties non-neg coefficients
Output: The non-existence of a positive root.
I often find myself in situations where I have an object with certain properties,
so a way of searching on theorems about objects with these properties would be very convenient. This could be expanded into a database where each theorem has a computer-verifiable proof.
A: John Rognes and I have posted a database containing  a minimal resolution of the mod 2 Steenrod algebra in the range $0 \le s \le 128$, $0 \le t \le 200$, together with chain maps for each cocycle in that range and for the squaring operation Sq^0 in the cohomology of the Steenrod algebra. The included document CohomA2.pdf explains the contents and usage of the dataset in detail.    From this data, all products in this range and all brackets of the form $\langle h_i, x, y \rangle$ can be computed, as explained in the document CohomA2.pdf.   See
https://archive.sigma2.no/pages/public/datasetDetail.jsf?id=10.11582/2022.00015
Also, note that all references to "www.math.wayne.edu/rrb" in the comments above should now be changed to "www.rrb.wayne.edu".   My tables of Ext charts there are wildly obsolete (but correct -- a wonderful thing about math is that it is never really obsolete).  When I posted them, computers were enough slower that these charts were possibly useful.   Now, it takes so little time to calculate Ext through t < conn(X) + 80 or so that it hardly seems worth posting Ext charts unless you also post information about differentials, or at least about chain maps which connect you to Ext charts where you do know differentials.
A sensible set of such would be a modern version of Mark's 1967 "Metastable homotopy" memoir, with all three homomorphisms of Ext induced by the maps in each cofiber sequence $RP^{n-1} \to RP^n \to S^n$, or better, $S^{-n} \to P_{-n} \to P_{-n+1}$, or even  $P^k_{-n} \to P_{-n} \to P_{k+1}$, where $P_i^j = RP^j/RP^{i-1}$ (extended by James periodicity) and $P_i = P_i^\infty$.
