Answer: we don't know yet, because there are Infinity ways to search. But for action=opensearch, http://ironholds.org/misc/api_presentation.html might make an interesting read.
We're trying to document all the various forms of API search so that we can do more ad-hoc and regular analysis of this stuff. There's an awesome start at https://www.mediawiki.org/wiki/API:Search_and_discovery that looks pretty complete - if anyone is aware of API methods /not/ listed there, please put them in, or reply here and I will :)