I've got flats on my P bass (and had them on all my P's) I keep trying flats on my jazz, even recently had EB cobalts which are very bright. I just can't get on with that sound on a jazz! To me a jazz should have that zingy hollow snap with both pickups on or that burpy punch on the back. I also love the slap sound on a jazz, and I haven't found any flats I like for slap.
That being said Joe Osborne and John Paul Jones always sounded fantastic with the jazz/flats combo.