Most names mentioned here are playing jazz, funk or fusion, but in the pop / rock genre there are some great bass players too.

Some that come to mind:

Mike Mills, from REM, the best 'melodic' bass player i know, especially on REM's earlier albums you can hear his brilliance.

Nick Seymour from Crowded House and Gary Love from Teenage Fanclub have a less unique style then Mike Mills, but are very gifted too.

Finally, a more allround player: Sonny Thompson from the NPG. I love the more sober, rockdriven work he has done with drummer Michael Bland.

Oh, i did forget another favourite:

Doug Pinnick from King's X: very powerfull, both funky and groovy.