I haven't noticed it being an issue with most newer op-amps, though it could be present on some. After working on mixing consoles I've gotten really careful about double checking to make sure that I can't lose a power rail, so even if it was present I'm not sure I'd notice it.
No guarantees here, but I think if an IC can function as either dual or single supply, it should be OK if it loses a rail.
Another factor would be the protection diodes from the input pins to the supplies. If those are overloaded or weak, that would be a failure mechanism. Especially for the op-amps receiving signal from off board.
Another factor would be the protection diodes from the input pins to the supplies. If those are overloaded or weak, that would be a failure mechanism. Especially for the op-amps receiving signal from off board.