Here are a few trainwreck covers (simply because I heard them recently):

Green Day - My Generation (The Who)
U2 - Neon Lights (Kraftwerk)
U2 - Dancing Queen (ABBA)

Basically, U2 should stay away from ruining other bands' music.

Though, to be fair, the instrumental work on "Neon Lights" is beautiful. But Bono's vocals are not to be believed.

Coldplay & Noel Gallagher
Stop Me If You Think You've Heard This One Before

Here are a few of my favorites...

AFI - My Michelle (Guns 'n Roses cover)
AFI - Just Like Heaven (The Cure cover)
Atreyu - You Give Love a Bad Name (Bon Jovi cover)
A7X - Walk (Pantera cover)
Bad Religion - We're a Happy Family (Ramones cover)
Bayside - A Long December (Counting Crows cover)
Bullet For My Valentine - Crazy Train (Ozzy cover)
Def Leppard - Ziggy Stardust (David Bowie cover)
Evergreen Terrace - The Kids Aren't Alright (Offspring cover)
Evergreen Terrace - Mad World (Tears for Fears cover)
Killswitch Engage - Holy Diver (Dio cover)
Marilyn Manson - Five to One (Doors cover)
Me First & The Gimme Gimmes - All My Lovin' (Beatles cover)
Metallica - 53rd And 3rd (Ramones cover)
Motörhead - God Save the Queen (Sex Pistols cover)
Murder City Devils - Hybrid Moments (Misfits cover)
Nirvana - The Man Who Sold The World (David Bowie cover)
Ramones - Have You Ever Seen the Rain (CCR cover)
Red Hot Chili Peppers - Havana Affair (Ramones cover)
Tiger Army - American Nightmare (Misfits cover)
The 69 Eyes - Call Me (Blondie cover)

Chris Martin AND Noel Gallagher, covering a song written by Morrissey?

Douchebag overload, I can't handle it.

Not even clicking.

Jeeez you guys arent very open minded, just be cause Noel is an ass and Chris Martin is in Coldplay doesn't make it a bad cover. Because 1. Chris is a good singer and 2. Noel & Jonny Buckland are good guitar players.

Gregorian does some superb covers, IMO, that is if you like that musical chant style. Personal fav is their cover of "In the Air Tonight".

Fiona Apple - Across The Universe
The Vines - Sorry Ms. Jackson
Smith - Baby It's You (from Death Proof - youtube this song...it's amazing)

Astonishing cover by Bettye LaVette of The Who's 'Love Reign O'er Me'.

He who has ears, let him hear...

Dinosaur Jr. - Just Like Heaven
How to take a great song and make it better.