Since I started taking piano lessons in the 3rd grade, I've had to perform in front of people occasionally. Recently, I've been doing four or five productions each year, so I think I could say I perform fairly often - probably somewhere between 25 and 50 times a year. Despite this, I still get anxious before nearly every performance. This past weekend, we closed a run of The Fantasticks, and a couple of hours before the show each night, I started running my lines and lyrics in my head. I didn't particularly want to - I had no major memory lapses during the whole run - and yet, I still have this lurking fear that my memory is going to completely fail, and so I compulsively run through everything I say and do, again and again, every night. The upside is, once the show has started and I've got the first song/monologue/whatever out of the way, the anxiety fades away and I can have fun.
I suspect different performers have different specific anxieties that manifest as "stage fright." Mine is specifically tied to memorization, and in the process of coping with it, I've learned one particularly valuable lesson - in order to have something (let's say a song, though this is just as applicable to dialog, choreography, etc.) "well-memorized", you don't need to be able to keep the whole song in your head at any given moment. I used to think I did, and while just starting to sing the first verse, would try to recall pieces of the later verses - if I couldn't, I would start to panic, and lose my place altogether. Fortunately, this usually happened in rehearsals, but I did it in a performance a couple of years ago (singing a duet from Phantom as part of a choir concert), and just managed to muddle through singing a mish-mash of verses. Amazingly, most of the audience members and other singers to whom I mentioned it after the performance didn't even notice, or at worst, had thought something might have gone wrong, but weren't sure what. The point is, audiences are incredibly forgiving. They frequently won't even notice minor mistakes, and are more likely to express sympathy for a performer who makes a major blunder than to heckle them. To get back around to the main topic, though, the most valuable thing I have learned is to have faith in your sequential memory. If you learn something well enough that you can do it correctly, at home and in rehearsals, in it's proper order, you know it will enough to perform it. No one cares if you can write out verse three while singing verse one, or vice versa.