I have always thought of the following covers as the “definitive” versions:
Superman - REM
Tainted Love - Soft Cell
I Love Rock & Roll - Joan Jett
I believe that Lou Reed stated at some point that he thought that The Cowboy Junkies cover of “Sweet Jane” was superior to VU’s original (or something to that effect)
Two others that rock:
Mr. Tamborine Man by the Birds.
“Gin and Juice” by Phish (Snoopy doggy dogg :P)
“Donna” by Clem Snide (Richie Valens)
“99 Red Baloons” Goldfinger (ehh…yea)
“Run Like Hell” by Kittle (Pink Floyd)
“Oi to the World” No Doubt (Vandals)
“Istanbul” They might be Giants (Four Tops?)
I know A LOT more better ones…just can’t think of em’ now…
Reoch
Hmmm…A few I disagree with ( for example The Zombies version of She’s Not There still trumps the otherwise pretty decent Santana cover, IMHO ), but a lot of good examples as well. A few more ( I can probably come up with a hundred of these 
)…
Another Replacements - Black Diamond ( originally by Kiss, a band I generally loathe ).
PJ Harvey’s OR Johnny Winter’s Highway 61 Revisited ( Dylan, of course ).
The Tremeloes’ Here Comes My Baby ( Cat Stevens - I like Yo La Tengo’s version as well ).
Speaking of perennial cover band Yo La Tengo ( I can think of several good examples in their repertoire ), I’m particularly fond of their version of the excellent Speeding Motorcycle ( Daniel Johnstone ).
ISiddiqui - What about William Shatner’s Mr. tambourine Man? Talk about a classic 
.
Soundgarden is “cock rock”? I guess I dont know what that means. I always figured they were one of the last harder “rock” bands with talent. Johnny Cash’s version is great, but not as good as the original IMO.
Pretty much anything by ** Me First and the Gimmie Gimmie’s** is better than the original. The best are “I Would Walk 500 Miles” and “Dont Cry for Me Argentina”
** NOFX’s** cover of Rancid’s “Olympia” (the whole NOFX/Rancid split has NOFX making Rancid sound like shit)is better, but how could it not be?
the Animals cover of “Boom Boom” was my favorite of all the versions I’ve heard of that song. I have no idea who’s song it is though and I’m to lazy to look right now.
dead0man
Ah, now you are talking ;).
That would be the late, great John Lee Hooker.
I was never much a fan of Sheryl Crow and I really detest Guns 'n Roses, but Sheryl’s cover of Sweet Child of Mine is one of my favourite songs. It’s just beautiful.
Also, I love Bob Dylan’s songs, but I hate when he sings, so any cover of Dylan is, to me, better than the original.
[secret shame]I prefer the Bangles’ version of Hazy Shade of Winter more than the Simon and Garfunkel original.[/secret shame] Don’t hurt me.
I admit I like the Limp Bizkit version of “Faith” better than George Michael’s. And the Alient Ant Farm version of “Smooth Criminal” better than Michael Jackson’s. But I don’t really like either of the original artists, so take that as you will.
I’d like to second, third, and fourth the endorsement of the JAMC’s “Who Do You Love”. It’s utterly brilliant.
Let’s see, there are so many…
The Rolling Stones’ version of “Not Fade Away” is a lot better than Buddy Holly’s.
The Beatles’ “Dizzy Miss Lizzy” was better than Little Richard’s and their “Rock and Roll Music” was better than Chuck Berry’s (as I often say- if you wanna remember John Lennon as a sensitive poet and artist, fine; I prefer to remember him as one of the all-time great rock and roll screamers).
Rod Stewart’s version of (I Know) I’m Losing You" is much better than the Temptations’.
For that matter, I preferred Rod’s version of “All Shook Up” (with the Jeff Beck Group, on the “Beck-Ola” album) to Elvis’ version.
Ritchie Blackmore’s Rainbow did a superb instrumental version of a really lame Yardbirds song called “Still I’m Sad.”
Nazareth’s rendition of “Love Hurts” was infinitely better than the earlier recordings by the Everly Brothers and Roy Orbison.
Aerosmith’s “Train Kept a Rolling” is better than earlier versions by the Yardbirds and Johnny/Dorsey Burnette.
any cover of a bruce springsteen song is bound to be better than the original, but “Blinded by the Light” and “Spirits in the Night” by Manfred Mann and his Earth Band are SOOOOOO much better than shortys.
you might also say that any cover Stevie Ray did was at least as good if not better than the original.
i really like Social Distortion’s “Ring of Fire” (not sayin’ its better, just that i really like it)
“Sick, Dude! Click it off! What the fuck is wrong with German people?”
To throw another Bob Dylan log on the fire I’ll nominate the Ministry version of Lay Lady Lay.
Also, the Clash’s version I Fought The Law (original by… Damn… Del Shannon?), Assemblage 23’s version of Love My Way (Psychedelic Furs), and Bauhaus’s Ziggy Stardust (oh, like you don’t know!) all top the originals.
The worst? Ever heard the (very very young and it isn’t even Robert Smith singing) Cure’s version of Foxy Lady? No? You lucky bastard!
I was going to say Steve Lawrence and Eydie Gormet’s version of “Black Hole Sun,” but I couldn’t keep a straight face…
More seriously, I would nominate any of the Who cover songs on LIVE AT LEEDS: “Shakin All Over,” “Young Man Blues,” and “Summertime Blues.”
And of course, The Troggs version of “Wild Thing” was technically a cover, although the original version is long since forgotten.
Bruce Springsteen- Chimes Of Freedom
Bruce Springsteen-Because The Night
Patti Smith-Because The Night
10,000 Maniacs-Because The Night
Yeah, yeah I know it might seem unfair to list both Bruce & Patti as covering Because The Night, but they co-wrote it so whenever they did it itcould be considered a cover, right?
The Monkees-I’m A Believer Their version is much more of a rocker then Neil Diamonds.
Pearl Jam-Crown Of Thorns taken from a few of their recent “bootleg” releases
David Bowie’s cover of “China Girl” is much better than Iggy Pop’s original rendition.
There is a special place in hell for people who believe this.
Of course, after saying that I must present my own “Are you joking?” offering: Pearl Jam’s cover of “Last Kiss”. Sure, it’s annoying, but it’s light years better than the famous oldie J. Frank Wilson and the Cavaliers version, which oddly enough was also a cover – the original was a local hit in Georgia for Wayne Cochran. I’ve never heard that version, nor the '70s cover by Canadian band Wednesday, but I suspect that the Pearl Jam version is the best of the sad lot. How is it that a song that wasn’t very good in the first place ended up being released as a cover three times and reaching #2 on the charts twice, more than 30 years apart?
I’ll go with:
Voices (Cheap Trick) Jon Brion
Smells Like Teen Spirit (Nirvana) Tori Amos
Suspicious Minds (Elvis) Fine Young Cannibals
Can’t Take My Eyes Off Of You (Frankie Valli) Lauryn Hill
Superstar (Carpenters) Sonic Youth
Silly Love Songs (Paul McCartney) Replicants
Powderfinger (Neil Young) Cowboy Junkies
Across The Universe (Beatles) Fiona Apple
My picks:
Tori Amos With A Little Help From My Friend
Quiet Riot Itchycoo Park. Man, I love this song.
Smashing Pumpkins My Blue Heaven
The Whitlams Tangled Up In Blue. Just to add more fuel to the Bob Dylan BBQ.