Ryan Adams, “Wonderwall”. (Oasis)
The Lemonheads, “My Name is Luka”. (Suzanne Vega)
I like a lot of Dave Edmunds songs that are covers. I like his “Girls Talk” better than Elvis Costello’s, and his “I Knew the Bride When She Used to Rock and Roll” better than Nick Lowe’s and “Queen of Hearts” better than whoever wrote the song. And his “Get Out of Denver” is better than Bob Seger’s.
Oh, and Louis Armstrong’s “Stardust” is the best version of Hoagy Carmichael’s song, ever.
Yes! That Sundays cover was one I was thinking of before I made this list, and then forgot about it by the time I was at a computer. U2 also did a cover of Wild Horses which I like better than the Rolling Stones original, but not as much as the Sundays cover. I agree about the “Painted Black” cover, too.
This version of Little Wing by The Corrs is one of over 30 covers of this original by… Jimmy Hendrix. 
Then today I was introduced to this version of Africa, the original being by Toto.
I liked José Feliciano’s cover of Light My Fire. I also like Roll Over, Beethoven by the Electric Light Orchestra.
Rodrigo y Gabriela did a lot of flamenco-inflected but otherwise spot-on covers early in their careers (and still do live), mostly Metallica but here’s Stairway.
The White Stripes version of “Jolene”.
The Disturbed version of “The Sound of Silence”
I Don’t Need No Doctor (Humble Pie/Rockin’ The Filmore)
I knew I’d forget one.
The Doubleclicks - “The Middle”. Jimmy Eat World does the song wrong, this is the way it’s meant to be done.
I much enjoy Save Ferris’ cover of Come On Eileen.
A lot of stuff by The Grateful Dead and Jerry Garcia Band, The Dead IMHO, were a GREAT cover band that also did original stuff.
Smokey Robinson’s Second That Emotion
I could go on and on!
You’ve never seen or heard a cuter version of “Single Ladies” than the one by Pomplamoose.
Leon Russell’scover of Jumping Jack Flash and Youngblood
Japan did a great cover of that cover.
Also, Faith No More have already been mentioned but I’d nominate them further for
and
Easy…urghhh!
Fucking genius.
Deep Purple’s cover of Joe South’s “Hush”
There was a CD a couple decades ago that was all covers of Saturday morning cartoon themes. Liz Phair can sing “Fleegle, Bingo, Drooper, and Snork” and make it sound sexy.
Most of them are pretty good, but the Reverend Horton Heat absolutely kills the Jonny Quest theme; 1:41 of scorched-earth guitar perfection. (Technically, the song then segues into “Hold That Pigeon”, but you can just stop in the middle.)
That’s probably most people’s pick, but my personal favourite would be the Was Not Was version, which has a couple of great rap interludes in the middle of it.
Props also to Gabrielle Aplin’s recent version of Franke Goes To Hollywood’s Power of Love
The Bonzo Dog Band’s demented version of The Monster Mashis far superior to the original.
The Allman Brothers Band’s version of “Statesboro Blues” is the definitive version. I’m also fond of Alice Stuart’s acoustic version.
Quicksilver Messenger Service turning Bo Diddly’s “Who Do You Love” into a 25-minute epic, and also do a great job with his “Mona.” Come to think of it, their rendition of “Happy Trails” is also pretty damn good.
David Bromberg does a great version of Mr. Bojangles.
Bado Pond’s epic, lysergic covers of Brian Eno’s Here Come the Warm Jets, and the Velvet Underground’s Ride Into the Sun.
Yo La Tengo’s multiple versions of Nuclear War, originally by Sun Ra.
The Cowboy Junkies doing VU’s Sweet Jane.
Starfucker - Girls Just Want to Have Fun
Foo Fighters - Down In the Park (Gary Newman)