Okay, gonna show my age here, big-time! 
Them for It’s All Over Now Baby Blue. I like it because from the way the song starts, you have no idea it’s gonna turn into IAONBB.
Your turn. 
Q
I like just about every Bob cover that the Grateful Dead did.
Well, this one is sooooo obscure, but my favorite Dylan cover is “Open the Door, Homer” by Thunderclap Newman. If you can find a copy of Hollywood Dream, snap it up immediately. Of course it contains Thunderclap Newman’s monster hit “Something in the Air,” but it also contains a beautiful cover of this little-known gem. (to the best of my knowledge, it’s only found on “The Basement Tapes”)
Close second is Leslie West’s cover of “This Wheel’s on Fire” off the first Mountain album.
As a rule, I don’t care for Dylan covers in comparison with the originals. There’s nothing more misguided than the old canard that Dylan was a great songwriter who should have gotten someone else to sing his songs. It may not be a coincidence that the two Dylan covers I can think of that I like better than the originals are primarily instrumental showcases that happen to feature singers who don’t have conventionally “good” voices: Jimi Hendrix’s “All Along the Watchtower” (that one’s a gimme) and the Nice’s Bached-up “Country Pie.” The Nice’s take on “She Belongs to Me” is also ultra-cool.
“All Along The Watch Tower” Jimi Hendrix
It’s not so much that Dylan couldn’t sing – he gives his songs a power that few others can match – but that he didn’t care about proper orchestration for the early songs.
In some ways that helps the covers because they can take what is almost the equivalent of a demo and put their own music to it and not sound like a bad version of the original.
So it’s hard to today to think of what the original of “All Along the Watchtower” even sounds like, so perfect is the Hendrix cover.
And even Dylan approved of the ways the Byrds expanded on his folk to create folk rock. In fact, there’s now a compilation record of these, The Byrds Play Dylan.
Those guys in The Band weren’t too bad, neither.
Holy Smoke! A little Googling came up with this insanely completist Body Dylan covers site. You gotta see it to believe it.
Body Dylan is evidently the twin brother that died at birth.
Shawn Colvin does a heart-rending cover of You’re Gonna Make Me Lonesome When You Go on her album Cover Girl.
Cowboy Junkies’ version of If You Gotta Go, Go Now (or else you gotta stay all night) and Odetta’s a capella Paths of Victory (both from a Dylan tribute album) are also fantastic.
Chris Smither performs an amazing Desolation Row and What Was it You Wanted? - this last one is also done wel by Willie Nelson
And Boots of Spanish Leather by the great bluegrass band Seldom Scene.
My personal favorite is Hard Rain’s A-Gonna Fall as performed by Edie Brickell and The New Bohemians.
I’m not a Dylan fan, but I really like Mike Ness’ version of “Don’t Think Twice”
Jon
That’s from a movie soundtrack; do you remember which one? (I’m thinking it was a war movie.)
I think it was Born on the 4th of July
Yes it was. I know it is on the soundtrack but am unsure if it was ever released on an EBATNB album. I suppose I could look it up but my gyros sandwich is calling…
I’m pretty darn sure it was never ever on one of their own albums. And a quick Google check seems to bear that out.
So Mindand is the one we’re listening to now, Exap’?
Q
I love Indigo Girls’ version of “Tangled Up in Blue” from their “1000 Curfews” live album. Especially the jam session toward the end.
Blackmore’s Night: "Times They are a Changin’
Well I’m not ashamed to say that I really like the Turtles’ rendition of “It Ain’t Me Babe”.
Granted, the Turtles were not a very infuential rock group of the 1960’s but their rendition has a lot of raw energy and power that I feel the Dylan version lacks.
Incidentally, the Turtles changed their angry musical style shortly after this song. Why ? Howie Kaylan said “We saw Barry Maguire’s ship sinking”. LOL
And no question, Jimi Hendrix’ “All Along The Watchtower” is a masterpiece.
That’s my favorite Dylan cover as well, but it is my favorite Dylan song overall. And I’m a HUGE Mike Ness fan.