These are the ones I came up with, and I'm pretty staggered there isn't one track from I Am Sam. Sarah McLachlan's Blackbird would make it Twenty-One. Hmph.
Anyway, here it is:
Arthur Alexander's Soldier Of Love - The Beatles
Bill Withers' Ain't No Sunshine - Tori Amos
Bob Dylan's To Make You Feel My Love - Billy Joel
Carly Simon's Nobody Does It Better - Radiohead
Cat Stevens' The First Cut Is The Deepest - Rod Stewart
Cat Stevens' Wild World - Mr. Big
Crowded House's Don't Dream It's Over - Sixpence None The Richer
David Bowie's The Man Who Sold The World - Nirvana
Don Henley's The End Of The Innocence - Bruce Hornsby
Don McLean's American Pie - Madonna
J. Frank Wilson and The Cavaliers' Last Kiss - Pearl Jam
Prince's Manic Monday - The Bangles
Simon and Garfunkel's Cecilia - Suggs
The Beatles' Come Together - Aerosmith
The Beatles' Something - Paul McCartney and Eric Clapton
The Beatles' You've Got to Hide Your Love Away - Oasis
The Chiffons' One Fine Day - Natalie Merchant
The Lover Speaks' No More I Love Yous - Annie Lennox
The Police's King Of Pain - Alanis Morrisette
The Supremes' You Can't Hurry Love - Phil Collins
Love them all, and I think every single one is better than the original. In my head at least, the gap between original and cover is easily mapped: Bob Dylan on the extreme left (read, miles ahead of the original), and the Beatles' on the extreme right (read, just as good, possibly a teeny, tiny bit better).
For more on covers, check out The Covers Project.