There's an option when asking a question were you can answer your own question, as you might know.
Is there a good example in this site of such a question/answer? How exactly should it be used?
I ran a query on the Stack Exchange API (code). Here are the 10 highest-scoring self-answers on questions from the past 5 years on Linguistics Stack Exchange (overall there were several hundret):
Obviously, don't abuse this functionality to farm reputation by repeately posting questions you already know the answer to. While self-answers along the lines of user6726's description may be useful for canonical answers to questions that are asked very frequently, Stack Exchange lives on the exchange between askers and answeres, rather than monologues of information written by a single person.
The prototypical use case of answering one's own question would be if over time, possibly by following up on the comments and answers by others, you managed to find an answer to your question on your own and would like to share the information for future readers.
This question has been discussed on meta: Can I answer my own questions, even if I knew the answer before asking?. The consensus seems to be that it's perfectly fine. In fact, if it is done with just one user account, it is less of a headache for me. As a moderator, some of my biggest headaches here are people who create multiple accounts, ask a trivial question from one account, answer from another account, and farm karma for both accounts. As long as it's all done with one account, I'm confident votes from other users will help promote worthwhile contributions.
If you see a better answer than your own, please accept that instead.
As for good examples here, I don't know. Some of the Q&As linked from the meta site show examples not related to linguistics, but at least demonstrate that good questions and answers accrue a lot of karma.
I don't know of any examples here, but I do know of them from Law SE. There function is to serve as a basic reference answer on a particular topic, and is usually tied to a tag. An example is "fair use", which people generally don't understand, and it is useful to have a technical summary question and answer. The "fair-use" wiki links to 4 fair-use questions and answers, one of which is a user who q'd and a'd her own question (it's the first q in the wiki), and it is a frequently-cited background source on the topic.
Such self-answers are good in the case of repeat questions, especially if users tend to ask very specific versions of the same basic question – an expert can write a general-enough question with answer so that many questions on that topic are thereby answered. It's not obvious to me that we get many questions that can be dealt with that way. For instance, we get a number of questions about "phonetic reasons for X", but there is no general answer, unless one plans to write a book on the topic (not appropriate for SE).