I'm only coming up with obvious stuff at the moment, so I apologize, but Sonic Youth and Velvet Underground are probably the two best examples this side of Fleetwood Mac. Then, y'know, there is
And Deerhoof, and Throbbing Gristle, and Yo La Tengo... Slowdive, My Bloody Valentine, Broken Social Scene, Arcade Fire, SWANS, Belle & Sebastian, Dead Can Dance, Fiery Furnaces, Thee Oh-Sees (sometimes), Low, Boredomes, New Pornographers, Blonde Redhead, The Knife...
Not giving any real thought to quality here. Just listing bands as they come to me, but I doubt there's a single band on that list that everyone's grandmother hasn't heard of.