there’s an episode of Grey’s Anatomy, where it was pretty much a musical episode (season 7, ep. 18). pretty much all the main cast members sang bits and pieces, some more than others.
This video shows the songs performed during the episode. Warning that, as this is a hospital drama, there will be scenes with blood and such.
Enormous actor Erland van lidthe de Jeude did NOT sing his song in the Gene Wilder/Richard Pryor prison movie Stir Crazy. I performed in college musicals with Erland, and am very familiar with his voice. I was shocked when I heard the singing, because it bery clearly wasn’t him. I later heard that there was some sort of union thing that kept them from using his actual voice.
Erland appeared in several other movies, but none of them featured his character singing until he appeared in the Stephen King/Arnold Schwartzeneggar film The Running Man. Unfortubately, there’s so much noise – explosions and lightning bolts – that you really can’t hear him. Also unfortunately, Erland died before the film’s release, so he didn’t get a chance to hear it.
Joan of Arcadia had a zombie musical. Hilarious!
Mel Brooks, title track from “High Anxiety.” Pretty good song, too – I enjoy breaking that one out every once in a while.
Someone mentioned Hoagy Carmichael above – his turn as Cricket in “To Have and Have Not” was cool.
Rita Hayworth, “Gilda,” and “Lady from Shanghai”
I don’t know if this counts, but Michael Parks and Bonnie Bedelia sang the title song for “Then Came Bronson” (Wayfaring Stranger is the tune, not sure if it’s Bedelia on it).
I do know that at least the keys on the soundtrack to The Committments were by a reasonably well-known performing musician. I think he lives in the bay area and is a younger guy.
“Rifle, Pony, Me” Rio Bravo
Too many set pieces for me to go on without feeling I’m missing the point of the OP, but it brings back some good memories.
You reminded me, that was the REASON I came in here, to post that, and got sidetracked completely. I thought he learned just enough to make the fingers look good, though – I hadn’t heard he actually is live on the audio track.
Otis Day and the Knights have toured as a real band after Animal House.
In Top Gun, Tom Cruise and Anthony Edwards sang “You’ve Lost That Lovin’ Feeling” in the bar scene, and were joined by Meg Ryan and Kelly McGillis on “Great Balls of Fire”.
I didn’t realise that when I first started reading it, and I was thinking how has no one mentioned Walk the Line yet. Then I looked at the posting dates - 2002. Walk the Line released 2005. Turns out I’d be very impressed if it had been mentioned 
Frank Zappa’s album, and intended Broadway show, Thing-Fish has a chorus of zombie Broadway stars.
9 years old? Ah well.
Great choice. That also ties directly into the OP since Glen Hansard is one of the Commitments.
Zooey Deschanel did her own singing in Elf, and would go on to form the band She & Him.
Juliet Lewis did her own singing in 1995’s Strange Days. Notably, she covers PJ Harvey’s “Rid of Me” in one scene.
Since this post was written in 2002, Guest, Shearer, and McKean also appeared in A Mighty Wind as the Folksmen. This was a skit they developed on Saturday Night Live in the 80s. Again, they wrote their own songs, played their own instruments and sang onscreen. In fact, Eugene Levy and Catherine O’Hara as Mitch and Mickey, also did their own performing, as did several other members of the cast.
Shearer, Guest, and McKean often appeared as the Folksmen, opening for themselves in Spinal Tap shows, and were usually booed, as concertgoers were unaware that both bands were made up of the same three men.
Fox’s playing and singing were not recorded in Back to the Future. Although Fox can play guitar and did play his own instrument in the movie so that it would look authentic, session musician Tim May played the actual guitar parts with Mark Campbell doing the vocal work on “Johnny B. Goode” and Paul Hanson, Fox’s guitar coach, played the instrumental audition piece near the beginning of the movie.
Robert Duval sang his own songs in Tender Mercies.
You can here Clint Eastwood sing in Honkytonk Man and Paint You Wagon.
Sylvester Stallone sang albeit poorly in Rhinestone.
Bruce Willis sings with Danny Aiello in Hudson Hawk.
And finally Natalie Wood did not sing in West Side Story, it was Mami Nixon, who also sang for Audrey Hepburn in My Fair Lady.
Natalie Wood did sing her own songs in Gypsy.
Salma Hayek has a song in the movie as we’ll
Gwyneth Paltrow and Huey Lewis in Duets. (I think that was mentioned before.)
Gwyneth Paltrow on Glee.
Steve Vai in Crossroads, of course plays his guitar parts. And Ralph Macchio’s.
Michelle Pfeiffer also did her own singing in Grease 2. And I’m pretty sure all the casts of Grease 1 and 2 sang their own parts.
Pfeiffer again, in Hairspray.
Mayim Bialik learned to play the harp for Big Bang Theory.
Antonio Banderas sings in Evita and has sung Phantom of the Opera on a TV appearance.
Hayworth was dubbed by Anita Ellis for both of those movies, although there appears to be some controversy over whether Hayworth is actually singing during the bar scene with the guitar in Gilda. I think she’s still being dubbed, though.
And James Caan is playing the piano himself during the “Auld Lang Syne” scene near the end.
I remember Steve Allen was asked on a TV talk show if he did any of the music when he played Benny Goodman in “The Benny Goodman Story”. He said that Goodman did all the clarinet parts except for one scene in the beginning when a young Goodman tries playing the clarinet for the first time. Apparently Goodman couldn’t make the clarinet “squeak” like a new student often does so Allen played that part.