It's a lot like an Ambitions profession in that the higher you get in your career, the more you can do. You start out doing Sing-O-Grams, and then as your sim gets levels in their career they unlock the abilities to perform at bigger venues. At level one, my sim has a single song she can perform, and she actually sounds good singing it, so I'm guessing you unlock more songs as you progress, or you might be able to buy songs at the bookstore (I haven't investigated that yet, so don't take my word for it).
Edit: No songs at the bookstore, so they much be are strictly career related.
Edit #2: The above is confirmed.