Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a short description.
I think that the tags are relevant, but feel free to change them.
Also, have there been any attempts to classify locally ringed spaces? Certainly, two large classes of locally ringed spaces are schemes and manifolds, but this still doesn't cover all locally ringed spaces.
In addition to the examples mentioned in the question, of manifolds and schemes, other commonly occuring types of locally ringed spaces are formal schemes and complex analytic spaces.
I don't know how extensive the taxonomy of locally ringed spaces is. For example, if $A$ is a local ring, we can form the locally ringed space consisting of a single point, with $A$ sitting on top of it. These are the topologically simplest locally ringed spaces (after the empty space). If $A$ is a field, one obtains a scheme. If $A$ is a complete local ring, one obtains a formal scheme. In general, this doesn't fit into any particular taxonomic grouping that I know of.
Incidentally, it might be worth mentioning that the various taxonomic classes can interact: for example, analytification of schemes over ${\mathbb C}$ is conveniently described in terms of maps (in the category of locally ringed spaces) to complex analytic spaces.
A locally ringed space is nothing but a local ring object (in the internal sense) in a category of sheaves over a topological space, which happens to be an example of a topos.
So in order to study general properties of locally ringed spaces, one could proceed by studying properties of local ring objects in arbitrary topoi. As any proposition (in the internal language) on a local ring (a commutative ring with $0 \neq 1$ and $s + t = 1 \implies s \in R^\times \lor t \in R^\times$) is true for any topos if and only if it can be derived intuitionistically (which is more or less the same as constructively), the original questions seems to boil down to:
"What are the constructively valid properties and constructions for a local ring?"
For example, the construction of Kähler differentials makes also sense constructively, which immediately implies (by the above reasoning) that every morphism $X \to Y$ of locally ringed spaces has an associated module $\Omega_{X/Y}$ of Kähler differentials.
And there is a lot of literature on constructive algebra. The book of Mines, Richman and Ruitenburg as well as many of the preprints on Fred Richman's homepage are a start. Some material can also be found in "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk.
An Introduction to Families, Deformations and Moduli, by T.E. Venkata Balaji, has a beautiful little appendix that introduces smooth manifolds, complex manifolds, schemes, and complex analytic spaces in a unified way as locally ringed spaces. Although it doesn't say much in general about arbitrary locally ringed spaces, I enjoyed reading it and seeing how the stuff I knew about particular classes of locally ringed spaces fit into the general framework. One thing that particularly struck me, although it's obvious in hindsight, was the remark (A.5.5) that for all the categories of spaces mentioned above, a morphism as defined classically is the same thing as a morphism of locally ringed spaces.