<$Xvarenah$> has a great list of favorite cover versions so I'd thought I'd list some of mine--

Walk Don't Run--Penguin Cafe Orchestra
Summertime Blues--T. Rex/Blue Cheer
Its All Over Now, Baby Blue--Them
It's Not Easy Being Green--Van Morrison
Reason to Believe--Rod Stewart (other great Tim Hardin covers include
Red Balloon--The Small Faces
How Can You Hang On to a Dream--Fleetwood Mac)
Who Knows Where the Time Goes--Nina Simone (used to great effect in John Malcovich's The Dancer Upstairs)
To Sir With Love--Al Green
Beginnings--Astrud Gilberto (like the Chicago version, with a longer trombone solo and percussion breakdown)
These Days--Jennifer Warnes (John Cale, who played on Nico's original, produced this version for her '72 Jennifer album)
Sister Ray--Joy Division (great, but worth it for Ian Curtis' "You should hear us do Louie Louie..."
Police on My Back--the Clash
Heartbreak Hotel--John Cale
At Last I Am Free--Robert Wyatt
Tomorrow is a Long Time--Elvis (Dylan's favorite Dylan cover)
Tomorrow Never Knows--801
Early Morning Rain--Jerry Lee Lewis (on his '72 London Sessions"unplugged", he puts you on that sad runway like Gordie never could)
Spoonful--Cream
Spooky--Lydia Lunch (great Robert Quine guitar on this)
Mary Tyler Moore Show Theme--Husker Du
Top of the World--Sugarcubes (I miss Einar)
Summertime--Billy Stewart (completely over the top)
Danny Boy--Jackie Wilson (and then some)
Gloria--Patti Smith
Move on Up--the Jam (the Curtis Mayfield version was a British hit--you flipped it over, like What I'd Say)
Honey Don't--the Beatles (who could sound like a Bakersfield bar band when they had to)
Layla--John Fahey
Are Your Ready For the Country?--Waylon Jennings
Different Drum--Linda Ronstadt
The Bells--Laura Nyro
Now I Wanna Be Your Dog--Alejandro Escovedo
Feel Like I'm Fixin to Die--Bob Dylan
Whole Lotta Love/Whiter Shade of Pale--King Curtis
Somewhere--Aretha Franklin
Some Other Time--Tony Bennett/Bill Evans
Trouble Man--Stanley Turrentine
Four Strong Winds--Neil Young
Work to Do--Average White Band
The Things We Did Last Summer--Lesley Gore
Come Down in Time--Lani Hall (from Brasil 66)
Hello It's Me--Isley Bros.
Only Love Can Break Your Heart--St. Etienne
A Walk on Gilded Splinters--Humble Pie/Johnny Jenkins
I Heard it Through the Grapevine--The Slits/CCR
Gimme Shelter--Mitch Ryder
Go Now--the Moody Blues