I take back that last sentence, looks like “in:solved” and “in:unsolved” are the terms to use to search for solved and unsolved items. So this is something admins / staff can use.
From an end user point it would be great to have a box/button that would hide solved posts leaving only unsolved or vice versa. I don’t see end users trying to search the discourse forum to figure out how to show only solved or unsolved posts
EDIT: A drop down list option in the advanced search would be nice, but a button / box on the categories page would be brilliant and intuitive.