Lots of great responses here. I'm glad to see Shellac and Jesus Lizard mentioned.
David Wm. Sims is one of my faves.
Japan also, Mick Karn RIP.
There are a lot of bands from the 80s and 90s I can think of that had real bass-heavy or bass-centered grooves. Some are rather obscure.
Hunters & Collectors from Australia. Especially up through the Jaws of Life album. Great bass playing.
Shriekback. Some of their work is pretty dated sounding now, but featured Dave Allen from Gang of Four (also bass-heavy and great band) so there are excellent, fnky, sinewy and sinister-sounding bass lines throughout.
Joy Division. Never listen to them anymore, but an influence back in the day. Peter Hook frequently carried the melody.
The Birthday Party. Nick cave's old band. Tracy Pew was great and setting a foundation for everyting else to fall apart over in a hot mess.
Obviously the Minutemen. Mike Watt. Come on.
Cocteau Twins had some great bass work. I especially like ther more raw early stuff with Will Heggie on bass, but a lot of their music is bass-fueled.
Finally, one of my all-time favorite bands, Killdozer. Michael Gerald was the bass player and also the lead singer. Pummeling, thudding, but groovy bass work. The band is hilarious, lyrically-speaking, but the music sounds threatening.
I realize that reading this list I could come across as someone stuck in a certain decade or mired in mostly punk and new wavey music, but I have very far-reaching tastes. These are just the bands that come to mind because they don't all get a lot of attention and these were the bands that I was copping from when I was really getting involved in playing bass all those years back.
Oh--honorable mention to Echo & The Bunnymen.