I think character studies are the sort of things you know when you see/read them, but which are very hard to define. I was afraid, fandom might yawn at the idea, assuming that smut was ruled out--but I don't think so since I can think of several pieces I'd called character studies that were pretty much porn from start to finish. But perhaps they're not 'shippy'? Again, not necessarily: there are cases where a character is explored in an interesting way within a ship context. See how tricky it is?!!
I guess the best thing I've come up with is that a character study illuminates an aspect of a character that I either wasn't aware of before or which isn't commonly explored. It casts a spotlight on a particular character and portrays them in a well-rounded fashion, not necessarily presenting them as a 'hero' character, but exploring their inner world, behaviour, emotions or relationships with others. Or something.
What do you think? The poll box is quite small so if your answer is complicated, you might prefer to comment. *g*