Let me give an example of question that I would consider a good candidate for closing as "good, but doesn't belong here": a question asking to write a good code in a specific programming language to solve a specific math program. Those questions will find a rich base of highly competent people to discuss it on Stack Overflow.

Here's a specific example which in my opinion is very well-suited for StackOverflow (where it was posted).

I was an active participant of Stack Overflow before I found Math Overflow, and I would really like to answer the above question. But I've come to realize that the best thing to do if such a question is posted on MO is to "transport" it somehow to SO. For example, it doesn't really fit the "arXiv+" classification scheme.

Again, my post was not intended to convey any more complicated thoughts than the observation above.

Thus, even a very well-written and interesting question is likely to get closed if its main topic is far enough from pure math to suggest that the experts in this particular question would be concentrated in other places.

The questions within the core competency of mathematicians, that is, mathematical questions, are sometimes vague, like, indeed, many questions about K-theory. But it's often immediately clear for the person who worked on K-theory what the question is, even if it is sloppily written, poorly thought or contains incorrect statements.

This is the way community works — and I don't think I have an easier way to formulate what types of questions will be welcomed by voters and moderators than "they should be interesting to mathematicians". Perhaps somebody will be able to explain it better than me.

@Andrew: One strategy I used early on to get people to start using MO is to find (or ask) a question that I think somebody knows the answer to and email them a link. Something like

Hi X,
Here's a question that I'd like to know the answer to for which you probably have an answer:
http://mathoverflow.net/questions/12345

This poses the central question very well. I am thinking about it, and I think I disagree.

Which bit are you referring to?

As mathoverflow's answer to Jon Skeet, I think that your opinions are very important, and explaining what you do is useful to the rest of us so thanks for doing so. Please carry on.

I should say, in case it's not obvious, that the "walking in the rain" question was partially an attempt to figure out what is a good question for MO. I've said this elsewhere but I think it worth repeating here to be sure everyone understands. That's one reason why I think that closing it was unfortunate - it means that it can't be used as a test case for the community. I also want to say that I have absolutely no issues with that question being voted down nor with anyone expressing (politely, as Tom has done) their opinions on it.

Ultimately, I'm still trying to figure out where MO fits in in my arsenal of mathematical resources. Not being an algebraic geometer, it's not proving a good fit in my research so my questions have more pedagogical or just plain "that's interesting, or is it?" motivation. This naturally puts them nearer the borderline but if it helps clarify said borderline then that's still a useful endeavour.

What I disliked about the question was that Andrew didn't present any evidence that there was anything mathematically interesting going on. (Indeed, he used the words "particularly dull".) It's very easy to ask questions of the form "Here's a real-world situation. Is there any interesting math behind it?" You could rattle off dozens of such questions. Whether there turns out to be interesting math behind this question is beside the point. – Tom Leinster

This poses the central question very well. I am thinking about it, and I think I disagree.

I downvote questions all the time because I know that the answer is elementary (example). On the same grounds, I downvote applied questions where I can see a solution using undergraduate math that I think should be obvious to a professional mathematician. (Not finding an example right now.) But, if I think about the question a little and don't get anywhere, I ignore it and wait to see what other people post. Running in the rain seems borderline to me.

I think that the right standard here is the same as for pure math: show that you have put some thought into the problem, usually by mentioning what you've already tried. In particular, it would be good to indicate you have some understanding of what a mathematical model looks like. I tried to do this when I asked my physics question.

I still think, though, that people are holding applied questions to a higher standard, and one that seems unreasonable to me. I'll see if I can give some examples in a bit.

I notice that Tom Leinster has expanded on his reaction. I have some sympathy for his point of view; I would argue, though, that I think that this particular subject comes up more often than most and that as professional mathematicians we should be aware of ways to sell our subject - but maybe that's an argument for a single meta-question rather than lots of little ones. I also tried quite hard to ensure that the question was focussed and had a possible answer that didn't involve going in to all the details of the mathematics of walking in the rain.

One slightly bizarre reason for reopening it would be to allow it to sink down in the morass. I think that the fact that it was closed has kept it further up the "active" list than if it had merely been answered and forgotten. If I'm reading the reputation scores correctly, then it is currently 11-4 in favour, with 2 favourites. I suspect that if it hadn't been closed, it wouldn't have gotten half of those votes.

More generally, I'm not in favour of closing questions that turn out afterwards to not have a good answer. I think that closing a question sends the message that this shouldn't have been asked. Slightly more borderline is the type of question that shouldn't have been asked in that way (the one on Badiou and Mathematics springs to mind); I would still vote to close these but with a "revise and resubmit" comment. But closing a question just because it turns out later to have been a daft question sends the wrong message, I think. I like to keep in mind the following quote about working for Pauli:

It was absolutely marvellous working for Pauli. You could ask him anything. There was no worry that he would think a particular question was stupid, since he thought all questions were stupid.

That said, I (almost) always leave a comment (or vote up a comment) explaining why I've downvoted or closed a question. If I can't put into words what's wrong with the question, then there is likely nothing wrong with it. I don't think I'm putting topics outside my areas of interest at very much of a disadvantage with this strategy. I agree that it's a shame that some areas are so poorly represented.

Traditionally mathematics is divided into two kinds: pure and applied. The relation of the two is delicate. The great applied mathematician Joe Keller's definition is: pure mathematics is a branch of applied mathematics. He meant that mathematics, beginning with Newton, was originally concerned with answering question in physics, it is only later that the tools and concepts used were elaborated into theories that took on lives of their own.

If we take that point of view seriously, if we disallow applied mathematics then pure mathematics goes too. But I am all for applying strict standards; either the question or any likely answers should contain real mathematics, and merely asking us to turn a vague non-mathematical question into a mathematical one is unacceptable. The questioner should be expected to do his own hard work. But I think it should be okay to ask, say, what models of granular flow generate interesting mathematics. Given the crowd that usually hangs out at MO, there may not be many useful answers, but that might change, and I don't think it's right to scare away the people who want to discuss such issues.