Such as U-571 with Bon Jovi. No big deal is made, he’s almost unrecognizable and turns in a fine job as opposed to …say…Memphis Belle with Harry Connick Jr. His acting was fine but the film sure made a big deal of having HARRY EFFING CONNICK JR.!!! “Hey, come sing. C’mon come sing!” “Well…okay…” (cue orchestra).
Levon Helm was primarily known as a singer/drummer, but had a decent acting career, starting with Coal Miner’s Daughter (haven’t seen it, don’t know if he sings or not). I have seen him in The Right Stuff (he plays Yeager’s friend and engineer, Jack Ridley, and is the narrator); doesn’t sing, but there is one tiny reference to his music career.
There are singers and musicians all through 'Round Midnight, and no big deal is made about it at all. Of course, they’re all playing singers and musicians, so maybe that’s not what you were looking for. 
Madonna in A League of Their Own.
John Denver in “Oh, God”.
Not a singer, but Flea in “The Big Lebowsky.”
Lee Ving, lead singer of FEAR has 30 credits on IMDB.
He was also in Back to the Future 2 & 3.
Come to think of it, Henry Rollins has been in a bunch of stuff, too.
Bass player, not singer, but Patrick “Putter” Smith as Mr. Kidd in Diamonds are Forever.
Harry Connick Jr. eventually got to the “no big deal” level with Copycat, and I didn’t even realize he was in Independence Day.
Bob Dylan, Kris Kristofferson, Johnny Cash and David Bowie. Maybe a big deal was made in some cases.
Tom Waits- Domino, Book of Eli, Mystery Men, probably other movies I’m not recalling at the moment.
I don’t remember anyone making too big a thing about Sting in Dune.
James Taylor and Dennis Wilson in Two Lane Blacktop.
Huey Lewis in Duets. I only became aware of this film through cable. There might have been a big deal because he was co-starring with Gwyneth Paltrow, but I don’t recall it.
He was in Coppola’s Dracula movie as Renfield.
Tim McGraw has been in The Blind Side and a few other movies with no special attention paid to his music career.
Speaking of country singers…Dwight Yoakum in Sling Blade.
And his nine-toed fellow Nihilist was singer Aimee Mann.
Lyle Lovett has acted in several movies and tv shows.
Cher became so well known as an actress in the eighties and nineties, I think people forgot she had started her career as a singer.