I found out recently that one can use tags in chat. They're given by the code
How does one see chatroom comments with a particular tag?
I'm guessing there'll be hidden gems to find, that's all.
One thing you can use to your advantage is that Search is looking at the HTML encoded version of a post, rather than what would be rendered.
Since every tag generates link of the form
site/tagged/tagname you can try to search for "tagged/tagname". Notice that this will also return all oneboxed questions posted in chat with this tag. But if the phrase is quite common this can eliminated some false positives. (As a typical use case I could imagine that users of some room would agree to use some tag for messages related to some specific topic - so that this makes searching for them easier.)
Since this is dependent on the way the Stack Exchange chat is implemented in the moment, there is no guarantee that this won't change in the future. (But I have not heard about some big changes regarding chat in the near future.)
Some examples: duplicates in CRUDE, functional-analysis in the main chatroom, new-tag in Tagging. (You can compare this with results when searching simply for duplicates, functional-analysis, new-tag in the respective rooms.)