I'd add Bette Midler's cover of "Chapel of Love," which totally kicks ass, I was playing in a band (Los Pathetics) and we did this version when a recently married couple arrived at a gig and got a $100 bill in the tip jar.
It plays for me when I click on it. Strangely I don't see a picture on it though, just black with a button to click to start the video. Not sure why.
As much as I like Lloyd Cole's original, Tori Amos' "Rattlesnakes" is my top choice. Runner-up: Dylan's "Going, Going, Gone" as performed by Bill Frisell, Robin Holcomb & Wayne Horvitz.
I looked it up...and yes I like this better than the original. That seems to be the criteria for me...it has to as good or better
Metallica's cover of Bob Seger's "Turn The Page". To spite the Metallic version having a completely different meaning, I like their version and the spin they put on the song was touching to me ..... and it kicked butt as well
The Wong Kar-wai film "Chungking Express" has this song by Faye Wong playing while she rummages through her paramour's apartment. It's originally by the Cranberries ("Dreams"), but this is the way I first heard it. The same film also is where I heard "California Dreamin'" for the thousandth time, and the thousandth-and-first and... but that's another story.