From the point of view of the general community I think the advantages of asking on MO is that the question and its answer are available on the internet for posterity.

From the point of view of the questioner the advantage is that you may get a good answer from an unexpected person. Furthermore, it seems plausible to me that you might get the answer faster on MO as more people will see it. The main disadvantages for the asker are that some people won't speculate as freely in public, and that the askees may not see the question on MO depending on timing (which may result in the question getting answered more slowly or not at all).

From the point of view of the potential answerers the main advantage is that you go to MO when you're in the mood to answer math questions, whereas you might be checking your email for lots of reasons. Furthermore, you're less likely to waste your time answering it, because you can see more easily if it's already been answered.

The primary drawback to asking on MathOverflow is that more people will know that some entity (and some might know that it is you) is asking the question. This could be a drawback in that someone else might consider it their question (because they have an answer that they are using as a major part of getting their degree), or it is a question that might be easily answered (which may lead to some personal embarrassment), or it is a question which others might consider bad form in asking ("Hey, I found this on my friend's desk, by the way, she works for the NSA and her computer password is grigori, and it seemed like a cool problem to ask..."). If you are imaginative enough, you can probably find other reasons for not making the question so public; all the ones I can think are along the lines of it would help some people to keep it private or semi-private.

There are more convoluted reasons for not making questions public, but they involve unfounded assumptions which may not hold between a grant asker and a grant provider. If your intentions are good and you know of no harm that can come from making the question public, go ahead and ask it on MathOverflow. (For this post, harm includes things like not following MathOverflow guidelines.)

Gerhard "Paranoia Can Fuel The Imagination" Paseman, 2011.08.06

...which suggests that I spend too much time on MO. Still, it is more productive to waste time on MO than to do most of my other activities.

Generally I ask a few experts before I post on MO, but that is mostly because it is more time consuming to frame a question properly for MO than for a specialist.

Bill Johnson makes a very good point when he says: "it is more time consuming to frame a question properly for MO than for a specialist." For a simple enough but technical question, it would be enough to discourage me from asking MO.

I have another one that's been rattling in my brain for the past couple of days: it's a terminology question. On the one hand, I feel like asking it on MO because I wonder if there are several terms used by different areas for the same concept. On the other hand, it doesn't seem like this question is serious enough to warrant the MO posterity.

This is an interesting thread. Here are a few observations: A question not directed to a particular expert but to the community is not as much of a direct demand on the expert's time, and leaves the expert to respond if the question is indeed interesting to him/her. This fact leads to more questions being carefully formulated and asked, which is good for mathematics in general. There have been a number of occasions where my questions have been cheerfully answered by people I wouldn't have dared to bother.

This aside, I have a stockpile of specialized questions that I'm pretty sure would be answerable by only a couple of experts. Aside from the benefit of recording these for posterity, I can't see why these wouldn't be better handled by a quick e-mail to the person who can answer them.

I guess that a good rule of thumb may be: If you know someone who has a 90% chance of answering your question, and wouldn't mind doing so, ask them first. If you don't have luck with this for whatever reason, THEN bring it to MO.

Thoughts?

I like to use something like Qiaochu's approach. I sometimes post a question on MO, and then email a link to the person I think might have an answer. This has a number of benefits:

1. posterity
2. MO potentially gets a new user
3. it puts less pressure on the expert, and sometimes I get an answer from an unexpected source
4. the process of writing the question up for MO often clarified the question (I've answered a number of my own question only after thinking for a while about how to pose the question on MO)
5. the math renders on MO, but not in an email
6. the expert has options for how to reply (sometimes they email me an answer instead of posting it on MO)

If I'm really worried about wasting their time, I wait a few days before sending the link. If I haven't gotten a good answer on MO after a few days, it suggests the question isn't an easy one.