A Fender Jazz bass for me every day. I have two MIMs that the necks are thin and fast but round enough at the back to fill the hand comfortably without feeling chunky. They both have Gotoh 201 bridges on, not because of the more mass/more sustain argument, but they limit sideways movement and raise the strings above the scratchplate a tad so I can dig in without hitting the scratchplate with my fingers as I play. Necks are shimmed to allow the action to be reasonably low without a lot of fret buzz. The placement of the neck pickup gives me something to hook my thumb on. These factors have made the Jazz the most comfortable and natural feeling bass for me to play. I have tried and owned Precisions, Ibanez Soundgears and Warwick Corvettes, but I always come back to the Jazz bass. On one I have SD Bassline Hots to give me more of a mids emphasis, and the other has the standard pickups which give me a nice vintage tone.
I tried some MIA and CIJ Jazzes at the time I bought the first MIM Jazz, but the MIM just felt nicer to me and the fret finishing was better, plus I was much happier to tinker with and modify a £400 bass rather than a £900 one!
It may not be the most obvious choice to use a Jazz to play heavy/thrash metal, but it does the job for me.