The Clash’s cover of “Pressure Drop” comes to mind.
I guess Richard Thompson’s cover of “Oops, I Did It Again” would qualify, although I didn’t realize it was a good song until he covered it.
The Beatles and The Rolling Stones did great covers early in their careers. The former’s “Money” and “You’ve Really Got a Hold on Me,” and the latter’s “It’s All Over Now” are a few of my favorites.
I won’t bother with Dylan covers; there are too many great ones to list.
This doesn’t really qualify because I don’t love the original artist, but… coincidentally, yesterday I turned on the radio and heard a tune I’d forgotten about for lo these many years: the Lords of the New Church’s fantastically over-the-top cover of “Like A Virgin”. Pure joy!!!
Tori Amos’ cover album Strange Little Girls I quite like, but especially the title track, which is my 2nd favourite Stranglers tune.
I’m quite partial to the French band Nouvelle Vague. Not to everyone’s taste, but their version of Love Will Tear Us Apart is catchy, although it’s not my favourite. That honour goes to Swans
On a related note, the Young Gods did a good album of Kurt Weil songs. September Song is great. (It’s no Longue Route, though)
And then there’s David Bowie doing the Pixies. How cool is that?