Dont know will some of you Americans have heard of the Stereophonics, a welsh group, but they do an excellant cover of The First Time Ever I Saw Your Face.
And there used to be an Irish girl group, Bellfire I think, who did a very good cover of the U2 song All I want is You
Oops! I Did It Again - Richard Thompson may have done it as a joke, but it’s comes over as really bitter and in-your-face.
I liked the London Philharmonic’s version of Kashmir.
As well as a most of Sarah Brightman’s covers.
Gray Matter plays a rockin’ version of I am the Walrus, complete with trippy ending.
I was also always partial to Phish’s cover of the Will Smith classic, Gettin’ Jiggy with It.
I would put in:
The Cowboy Junkies version of VU’s “Sweet Jane”
This has to be the definitive version.
The Feelies version of VU’s “What Goes On”
A looser version that seems to capture the spirit better.
R.E.M.'s version of VU’s “Femme Fatale”
Much more listenable without Nico. The best of the R.E.M. VU covers.
Well, if we’re talking about Phish, their cover of Snoop Dogg’s “Gin and Juice” is fantastic.
I also second the Revolting Cocks cover of “Do Ya Think I’m Sexy”, although you should probably include Ministry’s cover of “Lay Lady Lay”.
But then I think Faith No More has got to be the greatest cover band of all time. They took the cheesiest pop song and covered it out of nowhere in the middle of any concert. But the apotheosis must have been their cover of the Nestles song. “Sweet dreams you can’t resist. . .” I can’t tell you how many switches that flipped in my teenage mind.
Argh! I guess my previous three would be more in the vein of “probable” covers. Since my two current faves in the improbable category (“Hurt” and “Boyz in the Hood”) have already been spoken for, I will add:
Fine Young Cannibals’ version of the Buzzcocks’ “Ever Fallen in Love?”
Cash’s remake of Personal Jesus was damaged by the fact that Depeche Mode already put out a terrific acoustic version. I probably would have dug what Cash did a lot more if not for that.
How about Johnny Cash’s version of Rusty Cage? I think that was great.
<rant>
I need to comment on the string of covers of songs that have come out (as far as I can recall) since Lemonheads did “Mrs. Robinson”.
That would include AAF’s “Smooth Criminal”, whoever did “Come on Eileen” recently, whoever did “Boys of Summer” (Ataris?)
These all suck. They’re what the uncreative do with covers. They just play the same song with whatever instruments they happen to be holding at the time, and maybe speed it up a little (“but they changed ‘Deadhead sticker’ to ‘Black Flag sticker’”.).
“Boys of Summer” was a wistful, sort of melancholy tune originally. The ataris took everything out of it that was good and replaced with crap. All of these tunes are just marketing ploys, guaranteed to get a little airtime, and usually just highlight how bad these bads are because you have something to compare them to.
Believe this: If “Boys of Summer” was meant to be that uptempo with that sound from the electric guitars, Don Henley would have understood it loooong before The Ataris did.
</rant>
Hey, I have that album, and agree with you there – the Celtic-flavored “Behind Blue Eyes” is mucho cool. (I actually really like the a capella rendition of the “When my fist clenches, crack it open…” part.) 
T.A.T.U.'s cover of “How Soon is Now?” (originally by The Smiths) has absolutely no right on earth to be any good. Novelty teen cod-lesbian band who played the sodding Eurovision song contest covvering a Smiths track? That just has to be awful. When I first heard it I felt it treaded the thin line between disastrous and absolute genius.
I am now fully aware that it is indeed genius.
Hole did what I think is a good cover of Fleetwood Mac’s “Gold Dust Woman” for a movie soundtrack (can’t remember which film).
Radiohead has sometimes done a cover of “Nobody Does It Better” in live shows. I have a bootleg of it, and it’s a fantastic, rousing number. Lead singer Thom Yorke introduces it as “the sexiest song ever written” giving proper… um, props… to Carly.
I’m also big on the No Doubt cover of “It’s My Life” right now.
. . . I liked Tom Jones version of Prince’s “Kiss”
I like the Bangles’ version of Simon & Garfunkel’s Hazy Shade of Winter. Really shouldn’t work, but does, to me at least.