"My Sh*t's F****d Up" by Warren Zevon. It appears in his Life'll Kill Ya album, which was released just a few years before he died and is about his visit to the doctors'. It's both funny and sad at the same time.
I'm not sure a lot of the bands mentioned (The Rolling Stones, Madonna, Michael Jackson, Nirvana) should be classed as having cult followings because they're all so damn huge!
The first names that popped into my mind were Nick Drake, Morrissey & Warren Zevon.