Not a cover exactly, but Rhett Miller of Old 97s took the melody of Dylan’s “Desolation Row” and wrote new lyrics. They got Bob’s approval before recording it and Bob got half a writing credit.
Most days my favorite Bob Dylan cover is this amazing performance of the Man In The Long Black Coat by Steve Hackett.
Van the Man!
Yet I was like… “did he just sing amphetamine and pearls?”
So TIL that the original Dylan has the line, in “Just like a woman”
With her fog, her amphetamine and her pearls
ETA: I’ll try and get this in before the edit window: Has anyone covered “Talkin’ New York?” from his first record? He kinda sings it in what I reckon is a Woody Guthrie timber. I love the line about trying to play in Greenwich Village, “You sound like a hillbilly! We only want folk singers here.”
Indeed. A lot of us fans chipped in, and he made it somehow.
Another cover of Highway 61 Revisited, by Dave Alvin
Dylan was very proud of this cover.
Yes! You, me, and a bunch of others did some good.
According to second hand songs, no.
j
I loved one of the comments on that site to the effect that she not only sang beautifully and imitated Dylan well, but she did it with a squirming baby on her lap and a giant mug of water in her hand.
As for me, I prefer Dylan himself for his earlier works. Covers usually sound weaker and less memorable.
This just popped up on my YouTube feed. I’d forgotten all about this song.
Nick Cave and the Bad Seeds - Wanted Man
You already know 
OK, here’s mine. The Boss doing “Chimes of Freedom” in East Berlin, before the wall came down.
Thank you, thank you so much for posting this. I was moved to tears and cried like a baby (I’m writing this still with tears in my eyes). Of course I knew about Springsteen’s concert in East-Berlin and its importance, I even knew the short speech he gave in German before the song. But I never knew that he dared to play “Chimes Of Freedom”. It’s such a strong plea for freedom in a song like few others, and the line “And for each unharmful, gentle soul misplaced inside a jail” hit so close to home, there still were many people of that kind incarcerated in East Germany at the time Bruce sang the song.
ETA: and one year later, it all came down. Still unbelievable and surreal to me.
I love Dylan! Not everyone needs to sing like Freddie Mercury or Whitney Houston. Many of my favorite Dylan covers have been mentioned so I’ll go with this one that hasn’t.
Everything Is Broken - Kenny Wayne Shepherd Band
Just Like Thom Thumb’s Blues by James Lee Stanley. I didn’t know it was a Dylan song till I looked at the liner notes. Very good.
Since the obvious covers have been covered, I’ll say You ain’t goin nowhere from the Nitty Gritty Dirt Band compilation album ‘Will the Circle be Unbroken - Volume 2’ (performed with Roger McGuinn and Chris Hillman).
It’s a sentimental favorite for a specific reason, in addition to just being a very pleasant Bluegrassy version of the song: when I was first learning to play guitar, I wanted to be able to sing along as well, but it seemed impossible to be able to do both. A guitar-playing friend had the album on hand, and used that song as a good example song to practice playing and singing, because it’s so simple – verse and chorus are both the same 3 chords (G, Am, C). Of course, many Dylan songs are similarly simple, and he could have chosen many others, but that ended up being the first song I learned how to sing and strum along in time with.
Here’s an obscure one: Clothes Line Saga as played by The Roches. The video is obviously from an old VCR, but it is listenable.
If by chance you like the Roches, you are in for a treat! If you’ve never heard them before, be warned: they’re a bit quirky.