(pic from the Donders Institute)Just discovered this page called Unsolved problems in linguistics. It's a rather incomplete list, but a start. This is the sort of topic that could easily form the core of a very interesting conference debate. Linguistics remains a wide open field with competing theories and emerging methodologies, and the big questions remain dark and murky. However, this page claims that the origin of language is the major unsolved problem. I definitely disagree. The main goal of linguistics, as I would state it, is to figure out how language works in the brain (hence, that is our major unsolved problem). From that, most other questions can be answered (btw, see The Language Guy's take down of a recent report regarding the word most here). As our understanding of the brain improves, so will our understanding of language. I don't dispute that understanding the origin of language could be of use, but it is hardly the center of the linguistics world )I realize the Derek Bickerton might disagree).
NOTE: After Googleing the phrase "Unsolved Problems in Linguistics" I found a number of other sites dedicated to the same topic, including a Wikipedia page; however there is clear plagiarism/borrowing going on somewhere as there is word for word similarity between these sites; not sure who's cutting and pasting from whom. But you need only go to one site to see the same stuff.