Well, Alfred Molina (Dr. Octopus) has done musical theater, even getting a Tony nomination for playing Tevye in the Broadway revival of Fiddler on the Roof.
J. K. Simmons, who played J Johan Jameson in the first set of Spider-Man films, starred in a revival of Guys and Dolls on Broadway playing and singing the roll of Bennie Southstreet. He had a few other musical rolls as well.
Jeremy Renner was a member of a band, the Sons of Ben. He’s sang on the soundtracks of North Country, Love Comes to the Executioner, and The Assassination of Jesse James by the Coward Robert Ford. Here’s a video of him singing.
Jamie Foxx from The Amazing Spider-Man 2 has done a bit of singing.
Nicholas Hammond, Spider-Man in the 70s TV show, sang in the movie The Sound of Music.
Here’s a stretch: Bill Mumy was in the 1990 Captain America movie. He has released several records and has played with a couple of bands.
Also in Ghost Rider. You might be forgiven for blocking that out, although not that Hulk is much better. He plays… what he often plays.
Angel Salvadore. The superhero most commonly known as Angel (or Archangel for a period) was played by Ben Foster in X-Men 3.
Don’t sell him short. He did Fish Heads!
I haven’t seen this in a long, long time so don’t remember a thing, but Thomas Dolby did the soundtrack and had a small role in Howard the Duck…
Guy Pearce (Iron Man 3) put out an album last November. Kevin Bacon (First Class) has 6 with the Bacon Brothers.
This one confused me. I thought you meant the Dolph Lundgren version, which actually was a theatrical release… in Europe. This was the guitar guy in the Thomas Jane (or second) version.
I meant Ghost Rider, not Hulk. I get him confused with Kris Kristofferson for some reason.
No, you’re not misremembering; he’s in both. Just pointing out that some might want to forget Ghost Rider. Elliott was one of the best parts of it I think. He was Ross in the Eric Bana Hulk (2003); Edward Norton one (2008) had William Hurt as a less focal character.
Michael Chiklis (Ben Grimm in The Fantastic Four movies) has a band called MCB and they’ve released two singles.
Rob Zombie was a voice in GotG…
Has Ben recorded anything?
Which reminds me… Seth Green, who did the voice for Howard the Duck in GOTG, was in one of Weird Al’s videos, “White and Nerdy”. I don’t think playing the part of a member of “Dingos Ate My Baby” on BTVS counts, though.
Nick Blood (Agents of Shield) is in a band called Shaaark.
Oh, and Doc Ock’s wife was played by Donna Murphy, who’s starred in many Broadway musicals, and has won two Tony Awards for Best Actress in a Musical.
will.i.am, who played Wraith in X-Men Origins: Wolverine, was in this group called the Black-Eyed Peas.
Thanks for the input, all! I knew I could rely on you geeks. 
Ah! A couple of others I forgot…
•Jennifer Garner (Elektra) performed on a charity album. Kinda sweet…
•Christopher Lee (General Miguel—“Captain America II: Death Too Soon” [1978]) has a notable secondary music career…mostly metal.
•For an honorable mention…Nightcat! Marvel’s company-owned singer-superhero, created by Stan Lee and RCA, who actually got an album released! Which failed completely, and the singer now does internet porn and Troma movies!
He also cancelled the apocalypse.