Pulp, "Common People". Named the greatest Britpop song of all time by Rolling Stone. Later covered by William Shatner.
Alanis Morissette, "You Oughta Know". Spent 5 weeks at #1 on the Alternative charts. Later covered by Britney Spears.
2Pac, "California Love". Joe Cocker also knows how to party. No one covered this that I know of, although since there are at least 7 2Pac versions, covers aren't really needed.
Oasis, "Wonderwall". According to ChartMasters, Wonderwall is the most streamed pre-2000 song on Spotify. Later covered by Paul Anka.
Tricky, "Black Steel". In this case, we're looking at the cover version. The original, "Black Steel in the Hour of Chaos" by Public Enemy, is from It Takes a Nation of Millions to Hold Us Back".
PJ Harvey, "Down by the Water". Her biggest hit in the U.S. No covers I'm aware of, but it was featured on Beavis and Butthead.
Bruce Springsteen, "The Ghost of Tom Joad". To an extent, this video represents Bruce covering himself. The original was mostly acoustic. Later it was covered by Rage Against the Machine. Finally, Rage guitarist Tom Morello joined Bruce and the E Street Band for this version, which to my mind is easily the best.
Coolio, "Gangsta's Paradise". This is something of a cover version itself, given how heavily it samples Stevie Wonder's "Pastime Paradise". Wonder even gets a songwriting credit. According to the inescapable Wikipedia, there are no profanities, because Wonder wouldn't have it.
Jewel, "Who Will Save Your Soul". At last, my cover version conceit is defeated ... I don't think anyone ever covered this. So I'm left with this anecdote: in 1995, I saw Jewel open for the next act on this list.
Liz Phair, "Whip-Smart". She headlined a show I saw in 1995 that featured Jewel as the opening act. Borrows from Malcolm McLaren's "Double Dutch".