I bought myself a Squier Vintage Modified '70s Jazz Bass a couple of months ago and I like it it immensely. The only problem I really have with it is that I feel the stock pickups lack definition and don't really do it for me, so I'm looking to replace them.

I don't normally play bass, when I do I play in a Weezer cover band or in a hard rock/blues band with my dad, so I'm unsure of what to look for. I play through a Fender Bassman into a 2X15 cabinet.

I want a nice clean tone but something that can also handle some fuzz or overdrive well.

Any suggestions? I'm willing to go active as well, although this would be my first active instrument, guitar or bass guitar.
That's what I put in my vm Jazz. No more dynamic than usual jazz pups, but very articulate and powerful, you can get more midrange and it sounds less thin.

You might also look at quarter pounders--they are notorious for being very articulate or defined. I have the p-bass one on my MiM but haven't ever personally tried the jazz set.

You don't have to worry too much about your genre, basically anything that sounds decent will fit into your mix well, and any jazz pups will handle fuzz and od better than a p-pup (from my experience anyway).