Actually, the bikes on sidewalks thing is a grey area. I mean, it's not likely that the cops will actually say anything about it. In fact, I've always assumed they're kind of blind to what bicycles are doing anyway just because they're a low priority.
However, there are probably a few places where it is illegal to ride on the sidewalk in Austin. One place that I know of where it's posted "no bikes on the sidewalk" is along Lamar where the sidewalks are really narrow.
For example, there's this sign at 6th & Congress right outside of Waterloo Records. Another place where it might be illegal is along South Congress because of all the foot traffic there from the shops. Of course, I'm not sure if this sign outside of Waterloo Records was put there by the city, or if Waterloo Records paid for it.
https://www.google.com/maps/@30.2716...7i16384!8i8192
Really, I have no idea why anyone would want to ride on the sidewalk anyway. I hate doing it myself, and I only resort to it on streets where I really don't feel safe, but generally around downtown, on Lamar and Congress, I don't worry too much. I think the general rule should be that if you don't feel safe riding on the street because of hills and you can't keep up with the traffic or there are visibility/weather issues, then I would say it's ok to ride on the sidewalk. Otherwise, you shouldn't, and especially not if there are a lot of pedestrians.