Trad, swing, ragtime, bigband, bebop, fusion, smooth, and probably several others, and they are mostly sooo different that it's hard to lump them under one umbrella term, except for the fact that mostly it's one form that has morphed into the next. The jazz principle is brilliance, but there are only a few types and individual samples that I can listen to now.
These days (go on, shoot me) I'll listen to almost anything by Bob James and almost no other jazz at all. Why? Because on almost every track he's playing with somebody different. Why? Because everybody who is anybody in jazz wants to play with him.
I mentioned Tutu, Paul ... did you get into that one?