What are your experiences of handouts in mathematics lectures? There are many different styles of lecturing, and many different aspects that are blended together to give a whole "lecturing style".  That said, I'm particularly interested in hearing people's experiences with so-called "handouts".  At one extreme lie the lecturers who "dictate" a set of notes (usually not actual dictation, but by writing on a board) whilst at the other are lecturers who distribute complete lecture notes in advance.
As this is math overflow, I realise that it is extremely unlikely that it will be possible to answer the question "which is better", and I realise that this probably depends much more on other factors than just whether or not notes where given out or not, but to help me decide what to do then I'd like to hear people's experiences - both as lecturers and students.  If anyone can point me to actual research on this in the (mathematics education) literature then that would be an unlooked-for bonus.

(Minor edit: in light of the way that comments get displayed in "short form", I'd like to make it clear that the "Andrew" referred to in many of the comments is not the same "Andrew" who edited this question!  Unfortunately, if I put this remark in a comment - which is where it belongs - then it wouldn't be seen by those casually stumbling across this question and so those most prone to making that assumption!)
 A: I know at least one person who actively prohibits students from taking notes during lectures: Ole Hald. See for example here. Note: though the link only talks about upper division and graduate courses, I'm fairly sure I remember that he did it in at least one lower division Calculus course later on.
A: From the point of view of (mostly) a student (full disclosure: a physics student!) I vastly prefer to have handed out notes.  
I have three undergrad degrees: math, physics, and astronomy (each from different departments with distinct styles), and am now a physics grad student, so I've seen a number of styles coming from each of these areas.  And I have to say, one of the absolute worst styles is math without any handouts (beaten only by the style where the professor exclusively tells stories or does not show up to classes!).  
As Terry Tao says below (or, more likely, above, if you have this sorted by votes!) it can be very frustrating to spend hours being confused because of a simple mistake in the lecture, or a simple transcription mistake in my notes.  But more than this, one of the (IMHO) major failings in most of my undergrad math classes was that they focused largely on discussing proofs and technical details, and not at all on why anything was like it was.  
So, I could, e.g., come out of an analysis lecture being able to prove the uncountability of the reals with Cantor's diagonalization argument, but, since we went through it so fast and covering every detail, not having any understanding of why it worked, what the significant parts of the proof were, or why it is important.  Even though in retrospect there was nothing tricky or difficult at all about this argument, I remember having a great deal of trouble understanding this at the time.
However, in the (few) math classes I had that handed out notes with technical details carefully spelled out in them, and in class focused on discussing why things were like they were, in developing some intuition, and trying to modify assumptions and see how things changes, I had a greatly improved understanding (and ability to correctly do the homework and exams!).
It has always seemed far easier for me to go through the technical details myself, taking my time, than to go through the qualitative arguments myself in a field with which I am not familiar.
In fact, I found that these kinds of more qualitative lectures actually significantly increased my ability to carefully prove things, since I was able to better understand how the steps in a proof were linked together.  As opposed to understanding the technical details in each step, which are useful, but only useful to make a step careful after you are sure the step makes sense!
In the classes I've taught, and had enough control over that I have been able to do like I wanted (which, admittedly, is only one class, a baby-quantum field theory for undergrads class!) I typed up notes for my students, and found that the students who put extra time into understanding the class did substantially better than in classes where I have not been able to do this.
Students who did not put extra time outside of class into studying did not do any better, but claimed to like the notes better.  
So I've felt that it's a good idea to hand out notes, since, at the very least, it will reward the students who are working hard and putting extra time into the class.  Those students are the ones we want to continue in our fields, after all!
A: Without pre-typed lecture notes (or a textbook that is being followed reasonably closely), many students often feel pressured to copy down every scrap that the lecturer writes down, in case they are missing out on something that will be vitally important later.  This often comes at the cost of the student being able to comprehend what is going on in real-time.  A related problem is that without the backup of official notes or textbook, a single typo in lecture can lead to hours of confusion on the student's part when reviewing his or her transcribed notes afterwards.  (The problem is mitigated somewhat nowadays by the plethora of online mathematics resources, combined with modern search engines, but the situation is still less than ideal.) 
Note also that while the lecturer may know in advance which portions of the lecture are important enough to remember, and which ones are more trifling, many students will not be able to make the distinction in real time, and will thus have to record everything, leading to a sub-optimal allocation of the student's mental resources.
To me, the above dangers are worse than the opposite danger that the students are lulled into complacency by the existence of official lecture notes, and thus cease to pay attention to the class.  The latter problem can be fixed by a variety of means (e.g. making the classes more interactive or entertaining, or making the homework challenge the student beyond what is presented in the notes), and in any case is more a matter of the responsibilities of the student than of the lecturer.  The former problem is however difficult for the student to address by himself or herself (using third-party lecture notes, for instance, is usually a terrible solution).
Ideally, the existence of lecture notes should free up lecture time to focus on other aspects of the course (e.g. one could do a simple example in class, and refer to the notes for a more detailed example; or a heuristic proof with some details partially filled in, with the more technical details left to the notes; one can also present the more improvisational and free-form side of mathematics effectively in lecture, whereas the text medium is far superior for presenting the polished and structured side).  Using class time to mechanically repeat what is written in the notes or textbook is a waste, and reduces the lecturer to essentially being a fancy text-to-speech synthesiser (this is the dual problem to that of the student being reduced to essentially a fancy speech-to-text synthesiser); instead, lectures should complement and support, rather than replicate, text, and vice versa.
I discuss these issues more in my teaching statement,
http://www.math.ucla.edu/~tao/teaching.dvi
A: There is one more advantage to having notes in class that nobody has mentioned so far, perhaps because it applies more to graduate-level classes. I often find myself confused because some fact on the blackboard uses words that were defined a while ago, and that I forgot the precise meaning of. Also, sometimes it's hard to keep track of what bigger purpose does some technical lemma the lecturer is proving supposed to serve. When notes are available, I can often just look it up right there and then. 
For this reason, I often find it helpful to come to seminars with my computer, and look up the paper that the speaker is talking about online.
A: I would like to make a point that, it seems to me, has not been discussed yet. Regardless of my opinion on whether students can get more or less from a given lecture by having a set of lecture notes in advance, I think that this habit prevents them from improving their study skills.
I think that two good skills are lost in this way: taking good notes in an effective way and being able to study from books.
As for the first one, taking notes need not be a mere exercise of copying from the blackboard. When I was a student I used to take notes of what the the lecturer said AND what he wrote at the blackboard, even adding connections on my own. In a sense I rewrote the whole thing in my words in real time. This usually resulted in a far better understanding of the lecture itself. It also forced me to keep a constant attention and not lose the pace, and prompted me to guess proof in advance to save time. It also forced me to stop the lecturer and make questions when I could not follow. All in all, it just made me more active. It was a really good exercise, and student should be encouraged to doing so, within limits.
As for the second one, it is a different thing studying from a set of notes that contains exactly what you need to know, presented in the same way as the lecture, rather than having to browse through a book which possibily contains more (or different) material, with a different emphasis. Sooner or later the students will need to resort to books, and at that time having some experience will help them not getting lost.
So, while lecture notes on WHATEVER will help you learn WHATEVER, they may hinder you learning other things later. I think a reasonable compromise may be to give lecture notes at the end of each topic, even with a small delay.
A: I provide my typed lectures to the students via my website.  Looking at a nice, standard linear algebra course, I found that although many people told me they greatly appreciated the notes and used them extensively, they did not obviously improve the students' performance on tests or apparent retention of material from one lecture to the next.  Having taught this class quite a few times now, I would say that the large structure is much more influential in teaching effectiveness than the perfection of the lectures.  Sometimes the structure precludes effective lectures!
(The much more nonstandard thing on tropical geometry I once did was a different story, but that's because the notes were also the only text!)
A: When you take an literature class devoted to, say, Shakespeare, this does not mean that the instructor reads Shakespeare line-by-line during class sessions. Rather, the students read the plays on their own time, and class time is devoted to discussion, analysis, etc. 
I sometimes wonder if a similar model might work in a mathematics class. One might distribute lecture notes with basic definitions, simple examples, and easy propositions to be read by the students before lecture; the lecture time itself would be devoted to the proof of difficult theorems, higher-level insights, etc. 
A: In undergraduate lectures, Prof. Gowers didn't give out notes in advance; all was written on the blackboard (from memory!) Sadly I never saw his graduate lectures, but I think they were the same.
Prof. Korner, in both undergrad. and grad. lectures, gave out notes with only the theorems and definitions, and did the proofs in the lectures.
I personally really enjoyed both approaches. I'm sure most people who've seen them will agree that they are both great lecturers (but their methods could lead to disaster if you're not a great lecturer).
As a student, I thought lectures with complete notes given out in advance were not bad, but they just didn't fire up my enthusiasm in the same way. The mystery and anticipation added excitement!
Of course there's no right answer, and different students will probably prefer different methods; you can't please all students all of the time (even pleasing some students any of the time can be difficult!)
A: One basic observation, as a student.  A big reason for providing notes is if the class works out of more than one textbook (or none at all!) and you want to keep the narrative straight.  The professors that I've seen do this seem to be doing it in order to get to interesting material through a path that requires minimal prerequisites, which can require working through a different progression of topics than any given textbook has in mind.  
For example, Melrose teaches an undergrad functional analysis class at MIT which uses no measure theory; the class is based on a self-contained approach due to Mikusinski for defining the Lebesgue integral without defining Lebesgue measure.  The class was taught in 2009 out of several sets of online notes written by other people and in 2010 out of Melrose's own notes.  
At the other extreme, Munkres distributed only one set of notes when teaching undergrad topology, and that was a writeup of an alternate proof of Tychonoff's theorem.  His textbook is more than good enough to serve in lieu of notes!   
A: The idea of handing out lecture notes in advance, thus allowing class time to be used more creatively, is indeed appealing.  However, I clearly remember the classes where I learned the most in college.  They were not like that at all.  Rather, the professor wrote almost an entire textbook on the blackboard, which I dutifully transcribed.  The simultaneous engagement of eye, hand, and brain was somehow engrossing.  I cherished these notes and inscribed them on my memory.  Now, when I teach, I meticulously prepare notes for myself but don't hand out copies.  It is up to students to write their own editions, so to speak.  This seems old-school, but I've found that it works.  To be sure, the pace mustn't be so fast that students become mere stenographers.  They need time to think things over.  And care is needed to distinguish the essential from the more trifling (often simply by repeating the former a few times).  While it isn't and shouldn't be the only way to teach, I feel there is a place for a more formal lecturing style alongside the exciting, freewheeling approach advocated by Prof. Tao.
A: It is common in some subjects--but not (yet?) in mathematics--to have everything on Powerpoint, prepared in advance of the lecture, and available on-line.  Students bring their laptops to lecture (when they come to lecture) and follow along on the web copy.  This presupposes you (the lecturer) know in advance what will happen, of course.  Unexpected questions are not welcome.  And the OP wants to do it at the beginning of the semester!  
A: This last quarter I tried passing out a set of "skeleton" notes. This is nothing more than a list of definitions, theorems, sketches of proofs, and exercises.  The first definition is numbered, say by "1" and then the exercises that follow it are 1.1, 1.2 etc. Perhaps this is followed by Theorem 2, with exercises 2.1, 2.2, etc. No theorems are completely proved, only sketches are given. 
If a student wants to know if they understand a particular definition or theorem, they just work the exercises that follow the theorem. 
I like this because it gives a logical structure to the course. Moreover, in some sense it makes mathematics an active activity for the students. When I lectured from them, I would work relevant exercises for the students on the fly. There is no false sense of security from these notes as the students have to do work to "read" them.
Overall it seemed to work well.
A: My short answer is to give students the notes in advance, and if that discourages them from coming to class and making the most of it, then I am a poor lecturer.
I used to think that I was giving a good lecture if I was more useful to the students than an hour reading the book.  Now I think that I am giving a good lecture if I am more useful to the students than an hour reading my own set of notes.  This is to say, if I can be entirely replaced by my notes, and if giving students my notes in advance means that they will not come to class, then I am not doing things right.
Yes, I understand that I am setting a very high standard, one by which most lectures by most professors (and me included) are not that good, but I think that this is the standard to aim for nevertheless.  I like many of the things that Tao says in his teaching statement, in particular about how lectures are to complement, not reproduce or replace, the book or notes.  And yes, I admit that this is easier said than done.
A: The past few years I've prepared and handed out "notes" to my first year Calculus classes.  Each set of notes I leave incomplete to various degrees.  
For the first part of the notes for a particular section I'll type out ~90%  of the information, I leave "blanks" for the students to fill in.  I try to leave out words and symbols in a purposeful manner so when the students fills in that blank they have contributed a "key idea"; for instance when defining an increasing function I might let the students fill in the inequality between two expressions.  In this way they aren't wasting time copying down everything I say and do, but they are still involved in the note taking, and they are paying attention to my lecture to try and fill in those missing pieces.
In the middle section of each of my notes (here I'm usually doing basic/classic example problems, and proofs of theorems)  I'll usually type the problem we're working and add in some leading sub-questions or "hints" on how to do each step,  and I'll leave space on the page for them to write in the "work" of the problem.  Some students choose to fill in this part as I do the problem on the board, others just watch me and then try to re-create the work later on their own time.
The last part of my typical note sheet will have some additional problems and space to work them but often little to no prompting or hints.  These are often the problems that I'll do "as time permits" and are often harder/longer/more involved.  My students learn that I'd like them to be able to do these problems but realize that most of the time their tests and quizzes will have relatively few problems of this type.
Handing out notes like these I get to feel like I'm not "doing everything" for them, and yet at the end of the chapter/unit/class they have a nice concise "best of" version of their text to help them know what I think is important for them to study.
A: I haven't really made up my mind on this issue yet. I think writing up lecture notes beforehand has 2 big advantages over "winging it":
  1) It allows you to be much more detailed and thoughtful in presenting material in your own voice then a spontaneous lecture could be. It also allows you to catch mistakes much more readily before they pass into the students' hands. 
  2) It creates drafts for future courses that can later be modified,expanded or rewritten however you choose with the possibility of eventual publication. Or not. Of course,the problem with this approach is the lectures become much more "rigid" i.e. it's harder to add material or change things as you go and you rethink things or your perspective changes.   I think the best answer is a compromise: Write up careful drafts,but leave blank spaces on the notes for handwritten additions or revisions you come up with on the spur of the moment-especially those that result from student input. These changes can then be incorporated into the eternally evolving draft. 
A: Once upon a time, I taught a very non-standard, absolutely student-centered
multivariable calculus course just using handouts. You may find the details here: (2012), Moore and Less!, PRIMUS: Problems,Resources, and Issues in Mathematics Undergraduate Studies, 22:7, 509-524 : http://dx.doi.org/10.1080/10511970.2011.639337 
Here is the first paragraph of the paper: 
This is the story of a very non-standard multivariable calculus course. I think
it is worth hearing about since it surprised many of my mathematician col-
leagues. At first glance, it surprised them since it was a non-lecturing course
in which no black (or white) board, and no computer was ever used; yet, it
was a multivariable calculus course covering all the standard materials of such
a course. Thus, my colleagues’ first question was “so, what was the means
of communication?” Fortunately, the answer is very simple; just handouts.
My usage of handouts somehow differs from the two ways you have introduced. Thus, I put this as an indirect answer to your question to broaden the possible ways you may use handouts in the future.     
