That’s one of my favorite Bauhaus songs.
A couple covers of rap songs:
Nina Gordon (from Veruca Salt) - “Straight Outta Compton” (NWA)
Explicit Lyrics
Ben Folds - “Bitches Ain’t Shit” (Dr. Dre)
Explicit Lyrics
Really enjoyed that one. I saw the JB and Dead Kennedys live in Charlottesville some 30+ years ago. A fine show.
You gotta love the Dead Kennedys: “Viva Las Vegas”
And one can always count on the Residents for a good Elvis cover: “Jailhouse Rock”
You know, I had no idea that was a cover, so thanks.
Alien Ant Farm “Smooth Criminal” (originally by Michael Jackson)
Blind Guardian “The Wizard” (originally by Uriah Heep)
Ensiferum “Lady in Black” (originally by Uriah Heep)
Iced Earth “Shooting Star” (originally by Bad Company)
Nightwish “Over the Hills and Far Away” (originally by Gary Moore)
Ronnie James Dio “Welcome to My Nightmare” (originally by Alice Cooper)
This one’s loads of fun: The Zimmers “My Generation” (originally by The Who)
Chris Smither - Statesboro Blues (Blind Willie McTell)
Pixies - Head On (Jesus and Mary Chain - a fun video too)
Evan Dando and Syd Straw - For Shame of Doing Wrong (Richard Thompson)
Warren Haynes - One (U2)
See also: Me First and the Gimme Gimmes (exclusively a cover band, so not exactly analogous to Dickies)
One more: Punch Brothers - Just What I Needed (the Cars)
Cardigans - Sabbath Bloody Sabbath
NOMEANSNO - Forward To Death (A capella Dead Kennedy’s cover! - “That’s the way a snare sounds”)
Cat Power, Satisfaction.
Sundays, Wild Horses.
The amazing Stan Ridgway and 16 Tons.
Good one! The various artists album “RT - Songs By Richard Thompson” has a lot of excellent RT covers. I also like Bonnie Raitt’s cover of “When The Spell Is Broken” from that album.
Sturgill Simpson - “In Bloom” (Nirvana)
Toots and the Maytals Take Me Home Country Roads
First Aid Kit’s cover of Simon & Garfunkle’s America.
Watch it! It’s wonderful.
Me First & The Gimme Gimmes coveringHank. Complete with bagpipe! 
MFATGG also do a killer version of Nat King Cole and Judy Garland tunes, too!
Al Kooper’s 1976 cover of Gary Lewis & The Playboys 1965 recording of Al Kooper’s This Diamond Ring.
Note: Kooper wrote the song but didn't record it himself for 11 years. Also worth mentioning that aside from token overdubs by the Playboys, and Gary Lewis adding his weak but recognizable voice to prerecorded vocal tracks by Ron Hicklin, the '65 recording was performed by the Wrecking Crew. PS:Mostly forgotten??
Pixies do Warren Zevon.
Pixies and Warren Zevon, two of my all-time favorites.
Pixies cover of Neil Young’s “Winterlong.”
Tom Jones cover of Prince’s “Kiss”
https://www.youtube.com/watch?v=yuH1XDtN4rE
Stevie Wonder on The Beatles’ “We Can Work It Out”
https://www.youtube.com/watch?v=cl3dN1L1XQE
Stevie Ray Vaughn doing Stevie Wonder’s “Superstition”
https://www.youtube.com/watch?v=iTUYKHOdHCc
Perhaps my favorite… Richard Thompson cover of Britney Spear’s “Oops I Did It Again”
https://www.youtube.com/watch?v=V4WGsMplGxU
Your post has led me on a tour of YouTube videos of Bruce and the band playing covers in concert, mostly during the ‘audience choice’ portion of the set that they do during every show.
This exercise has convinced me that the E Street Band is hands down the greatest cover band that has ever lived. As evidence, I present this clip of them doing Van Morrison’s Brown Eyed Girl. Bear in mind that the band has never recorded the song, they didn’t rehearse the song, and they didn’t know they would be playing the song until just a couple of minutes before they started…Bruce even goes back and forth a couple times trying to figure out what key they should play it in.
Emiliana Torrini I Really Loved Harold
Same artist The Man With the Golden Gun