As a performer myself, I can tell you that being on-camera is very different from being on *stage.* Even though a sitcom is taped in front of a live audience, you still have a chance to re-do something if you flub up. The audience members -- who are there by invitation or lottery, they don't pay -- understand that they're watching a "work in progress" that will be different from the final product. On stage, though, you don't get any "do-overs" -- you have to keep going. And since the audience has paid admission, performers feel the obligation to put on as polished a performance as possible.
Singing is worse, somehow. I guess it's because you're not playing a character; it's *you* doing something that's rather technical and not part of ordinary everyday behavior. ...I was once at an audition for a musical. One fellow auditioning was a seasoned veteran of the dramatic theater. He had played Othello and many other large parts. I mean, this guy knew his stuff. But I was sitting close enough to him during his singing audition to see that his hands were shaking like leaves.
Actors learn to use their stage fright to energize their performance. In fact, the time to worry is when you *don't* have the butterflies before the opening curtain. (I'm a piece of meat until I hear that first laugh from the audience; then I feel much better.) But I imagine that for singing, which requires more physical control, it could be a real problem if the nervousness gets out of hand.
My advice: - Do something physical before you go on. Deep breathing (don't hyperventilate), swing your arms around or just jump up and down a bit -- something to burn off some of the adrenalin. - Have something to "ground yourself" with when on stage. I know why so many singers lean or hold onto the piano; the first time I sang in public I dug grooves in the damn thing. Or have something in your hand; I've seen a lot of singers holding a handkerchief, or the microphone stand. - Acting is helpful in singing: Every song tells a story, so think of yourself as playing a character who happens to be singing his monologue. - Remember that the audience is really on your side. They *want* you to be fabulous. The Apollo Theater notwithstanding, nobody is going to boo and throw tomatoes at you. What kind of nut would bring vegetables to the theater, anyway? - I recommend improv classes to everyone who asks about any subject. It may not seem that improv has anything to do with singing, but trust me. Once you have some improv under your belt, nothing will ever scare you again. You'll be bulletproof.
...If all else fails, imagine them in their underwear. On second thought, don't; that's kinda why the last guy got fired. ;o>