It's a difficult question to answer. When listening to jazz, I'm usually focused on the instrument soloing/improvising at the time. More often than not, I'm listening to the difficulty of the piece - thinking to myself, "Can I play that?". So execution, rhythm, solo/improvisational ability, etc. In a more general sense, when I'm listening to jam bands, particularly Medeski Martin and Wood, The Disco Biscuits, the Steve Kimock Band, and Jacob Fred Jazz Odyssey, I'm listening to the drums, being a drummer, with the keyboards being next, being a keyboardist. If a harmonica part shows up, I'll listen to that pretty intentively, being a harmonica player
For some reason, I could never follow the bass parts extensively unless they're really prominent (think "So What" prominent). I listen to harmony often if it is pertinent, such as in big band pieces and classical pieces - in that sense, the viruosity of the composition in general is focused on.
What a loaded question/response!