I can think of one character that does this really well and I have a blast every time that character performs. Using an off-site theme-inspired playlist helps a lot, but it can also be tricky. Staying theme inspired and away from our own timeline's memes and cultural references takes a lot of attention to detail when setting something like this up, but i find it usually adds a lot more to the immersive atmosphere than it takes away from it. The player can really melt into the scene with the right musical choices and find a new sense of IC because of it.
That said, it isn't necessary. I've seen characters perform dance routines without off-site music that were just as compelling. I saw a bartender once perform her trickery behind the bar, too, and that was fantastic. I think the best thing we can consider when performing in any way on a stage in front of thousands (including ambient obviously) is that we are essentially ambient population ourselves. We aren't interacting with everyone in the room we're a part of the room. Posing and emoting with that in mind (ie. filling the room with beats or riffs or whatever, the general crowd's reactions, the lights, the atmosphere) while maybe singling out one pc in the crowd every now and then to connect to it in a more personal manner are all great ways to fill the musician's role. Its not just strumming and singing, though that has to be part of it, but it can be mixed into the scene while enhancing the atmosphere for everybody.
Those are my thoughts. I'm sure not everybody agrees. I wish there were more musician, though. My character looks for people to fill that role (though, my time constraints severely hinder what time i have to spend on something like that and that seems to fall to the back burner) but if musicians were actively pursuing clubs and bars to sign them up for a night it would be a no brainer. Having more entertainment means more gathering places and more competition, which means more diverse RP, which means more fun. :-)