If the light is a red circle, then turning right on red is allowed unless there are signs that prohibit the movement.
If the light is a red arrow, then turning right on red is NOT ALLOWED unless there are signs that specifically allow the movement.
These rules come from the Federal Highway Administration Manual on Uniform Traffic Control Devices. Only ten states have not adopted the MUTCD. The other 40 are split with just under half adopting the MUTCD as-is, the others supplementing the MUTCD with their own rules.
For me, the take-home lesson is to exercise caution when traveling. It may make someone mad if I don’t turn right on red, but there’s nothing saying I have to turn on red. If I’m not familiar with the local laws, I think I’ll just play it safe and wait for the green.