This is my all-time favourite cover - I've always loved Cohen and 'Pump up the volume' was the first time I heard Concrete Blonde and this version just blew me away. Without that movie and this song I would probably never have found Concrete Blonde at all and would have missed what is now one of my favourite bands. Their first 5 albums are all really, really good.
Since that one has already been posted though, I'll go with one of my other, and more recent, favorite Cohen-covers:
A cover of Little Wing by Eric Clapton and Sheryl Crowe. Have to give it them for pulling off a good one here:
It can take me a while to build up the courage to listen to some of these covers, and this one made me somewhat trepidatious, but it was really worth listening to. It's hard to cover Hendrix and Clapton acquits himself well