Ice T does a lot of straight acting work that doesn’t reference his career as a rapper.
Ice Cube, too.
He was actually scary in that. “Copycat” that is.
Chris Isaak in “Silence Of The Lambs”
Frank Sinatra in a whole lotta stuff.
I should have said I was looking for supporting roles and not leads or smarmy cameos. Waits in Dracula, Issak in Silence of the Lambs. Those are definitely two good examples.
Jani Lane from the band Warrant was in the movie High Strung and did an excellent job playing Vol and did a great job. It was a supporting role.
Lenny Kravitz played Cinna, the person responsible for Katniss Everdeen’s wardrobe and presentation, in both Hunger Games movies.
Mariah Carey in whateveritscalled Precious? Pushing Sapphires?
Every review I read mentioned her, but always expressing surprise at the quiet casting, unrecognizable makeup, and unexpectedly good performance.
Good one!
Sam Phillips played Katya, the mute female terrorist in the third Die Hard movie.
Helm didn’t sing in CMD; he was exceptional as LL’s father.
Was going to mention John Mellencamp until I read OP was looking for supporting roles. Doesn’t matter, his movie was awful.
A: It’s spelled “Lebowski”. Sheesh.
B: As punishment, I am going to go off topic slightly and list a few other actors in The Big Lebowski who have also sung: Ben Gazzara in Tales of Ordinary Madness and Husbands. Sam Elliott in Barnyard sings “I Won’t Back Down”. Dom Irrera in Hollywood Shuffle sang “Rebels of the Night”.
John Turturro was mostly dubbed in O Brother, Where Art Thou? but he did sing “You Are My Sunshine”. Jimmie Dale Gilmore (“Smokey”) was a semi-well known country-western singer. Peter Stormare sings in a few films including Dancer in the Dark- “I’VE SEEN IT ALL”. Philip Seymour Hoffman sings a couple songs in The Master. Julianne Moore plays a singer in What Maisie Knew and does a few songs as well as a song in the Carrie remake.
And most importantly John Goodman in many films including Blues Brothers 2000, Raising Arizona (“She’ll Be Coming Round the Mountain”), True Stories, and in Monsters, Inc. he sang the Oscar winning song “If I Didn’t Have You”. His singing talents helped get him several of his early acting gigs such as True Stories and Sweet Dreams (although he did not sing in the latter).
All of which makes Jeff Bridges look like a piker. What did he ever do musically? Besides winning a freakin’ Oscar playing a country-western singer in Crazy Heart. The Fabulous Baker Boys obviously doesn’t count. 
And his first real credit is an appearance with his dad on Sea Hunt in 1958 where he sung “Oh! Susanna”.
I now return you to your regularly scheduled thread. And you know Smokey has emotional problems.
Tim McGraw was surprising good in Friday Night Lights.
Alanis Morissette had a small role in Dogma. She didn’t sing, which, under the circumstances, may have been for the best.
Gene Kelly (admittedly more famous for his dancing than singing, rain notwithstanding) had a supporting role in Inherit the Wind (1960).
He was pretty smarmy, though.
Bette Midler. I’m not sure which career came first, but nobody in either field thinks she’s a gimmick.
going old school.
Dick Powell did a whole bunch of stuff.
Ken Curtis was an tv cowboy. he had replaced Frank Sinatra in the Tommy Dorsey band.
Ed Ames was a tv Indian.
Tony Bennett played a forgettable part in the forgettable movie The Oscar.
Trini Lopez in The Dirty Dozen.
Gavin Rossdale in Constantine and The Bling Ring.
Steve Van Zandt is best known as a guitarist, but he did do some solo albums before appearing in a recurring role in The Sopranos