Oh man… far too many to mention. Here’s off the top of my head:
Reel Big Fish - Hungry Like the Wolf (Duran Duran)
Revolting Cocks - D’Ya Think I’m Sexy (Rod Stewart)
Pain Station - Safety Dance (Men Without Hats)
Wesley Willis Fiasco - Girls on Film (Duran Duran)
The Cure - Foxy Lady (Jimi Hendrix)
Goldfinger - Rio (Duran Duran)
Fountains of Wayne - Baby One More Time (Britney Spears)
Brandywine Valley Green Grass All-Stars - Paradise Certy (Guns ‘n’ Roses… “where the grass is greens and the girls are purdy”)
Dead Kennedys - I Fought the Law (Sonny Collins), Viva Las Vegas (Elvis), Theme from Rawhide, Take This Job & Shove It (Johnny Paycheck)
Obadiah Parker, which is the band’s name, not the guy. I’m also a fan. Somewhere out there is the Outkast video synced with the Obadiah Parker music, and it’s kind of cool.
The Slits’ cover of “I Heard It Through the Grapevine.” It’s not played entirely for laughs like some others listed in this thread, but it is definitely off-kilter and somewhat demented, especially Ari Up’s amusing failure to switch the gender pronouns.
That would be the Circle Jerks’ “Golden Shower of Hits.” I think they do a cover of the Soft Boys’ “I Wanna Destroy You” on the same album, with Debbie Gibson singing backup.
If we’re looking for wonderfully weird, the Puppini Sisters are it. Actually, whether they’re actually weird or not, I don’t care…I just wanted to tout them!
They’re a swing revival group who do authentic 40s music as well as covers of more modern songs in 40s format. If you go to their page, you can cycle through about four songs, two of which, I think, are covers of Gloria Gaynor’s “I Will Survive” and Kate Bush’s “Wuthering Heights” (and if covering Kate Bush in swing/boogie-woogie style isn’t weird (although a REALLY good weird), I want you to tell me what is!)
Edited because Ms. Bush sang about more than one Wuthering Height…
All their covers were twisted (I think you got them all), but this one’s the best. They turned it into a commentary on the murder of Harvey Milk, and changed the chorus to “I fought the law and I won.”