The following is a draft [Added: please see my other answer for the current version, the draft itself here is obsolete] for a contribution to the 'What about open problems?' of the revised documention of MO. My aim was to interpolate somewhat beteween the two points of view given in the two answers, yet giving somewhat more weight to the one that had more support.
The intent was also to stress more what can be asked or done than what cannot, not only but also as it should be rather on the 'what can is ask' page than on the 'don't ask' one. Not sure how well I suceeded though.
I welcome feedback of all kind. Both regarding content, form, and details like the language (the goal is this will make it into the official documentation in one form or another so thus seems relevant). In particular, for the latter feel free to edit directly. (Also, for the former if you find it more convenient, but perhaps please leave a comment what you changed, so that it is a bit clearer than just from the revision history who thinks what).
To ask questions related to a (well-known) open problem, that is a problem of which you know that it (yet not its solution) is known at least to the experts in the relevant field, can be welcome. However, there are some things to consider, which are mainly consequence of general principles making a good question or a good problem a good fit for this site.
If you know that something is a (well-known) open problem, please, indicate this clearly in the question itself and provide some information or pointers to information on the context and status of the problem. In addition, please, use the open-problem tag (in addition to mathematical tags to indicate the subject). Otherwise, the unfortunate situation can arise that the only reply you get is something along the lines: 'This is a well-known open problem, see the following references.' This could be a valuable answer for you, had you not known it already, but since you know it already, it is likely not what you hoped for. Conversely, it can be frustrating for the person answering to spend time to provide you with information of which it then turns out you had already. Thus, please provide context.
This site is a question and answer site. Thus, the typical question should be of a form where you consider it as reasonably likely that it is answerable by somebody (at the time the question is asked). Therefore, this site is typically not a good fit for asking for solutions of a problem open since some time and known to most people most likely to answer it. Thus, if you know an open problem you consider as particularly interesting and you feel it is not sufficiently present in the literature or on the internet you might consider to contribute it on a site that collects open problems in mathematics such as the Open Problem Garden instead of asking it here. Or, you ask a related question that seems more answerable, such as a question on the current progress towards the problem, a question on known results in special cases, etc. (of course, always assuming that this information is both of interest to you and not readily accessible elsewhere; following the two general principles that you should try to ask only questions whose potential answers are of actual interest and value to you, and that you should try to look-up other sources of information before asking a question here).
A situation where you come across a problem that you, and possibly some colleagues around you, cannot answer but of which it is not known to you that it is also known to many others already is not what is meant by open problem here. A question on it is typically very welcome, provided it is presented in context and it is within the scope of the site (research-level mathematics) from a general point of you.
Added: the intent is to replace this text:
What about open problems?
MathOverflow is not the right place to ask open problems. You should post questions you're actually seriously thinking about. If you're thinking about a well-known open problem, provide some background and ask about something specific related to the problem, like "Such and such is a well-known open problem. So-and-so proposed this and that approach in the 80s. Does anybody know if this aspect of their proposal can be made to work under these circumstances?" If you want to contribute to (or view) a list of open problems, visit the Open Problem Garden.
If it turns out that a problem is equivalent to a known open problem, then the [open-problem] tag is added, and the question is converted to community wiki. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?" That way, the MO thread for the problem becomes a repository of resources related to the problem. Perhaps the answers could be organized by approach, with an outline of the basic approach, followed by a horizontal rule and a summary of what is promising about the approach and why it doesn't give a complete solution.
To join the discussion about how MathOverflow should deal with open problems, go to this meta.MO thread.
This is what the faqs said all the time (or at least since a very long time), and what also now is a part of http://mathoverflow.net/help/on-topic
In particular, this text is not a my opinion on how open problems should be handled but by contrast my attempt to write down how they are/will handled in accordance with the opinions expressed in other answers as well as in continuity with existing practise (as I see it), and also removing mention of procedures related to open problems until now mentioned in the faqs that did IIRC hardly ever happen like this in the recent and not so recent past (for example, I recall numerous questions where it turned out this is a well known open problem, but never was there any CW-ing or any attempt of creating anything like a resource).