One common test is the famous "mirror test" where an animal is given some problem that can only be solved by using its own reflection in a mirror for reference, such as a study involving an Australian ant. They put a blue dot made of felt (I think) on the ant's head behind the antennae, and watched the ant clean itself once it saw that it's reflection had a weird blue thing on its head. But I don't know if there are other tests for "self awareness"