What is the crime Oedipus Rex is guilty of?
This previous answer gave two crimes that Oedipus committed. In fact, murder and incest are what we moderns think of when we hear the name Oedipus. However, if we consider what the ancients thought, they may have come to a slightly different answer. They may have pointed to something more basic; we can even say this crime was the very thing that caused Oedipus to commit murder and engage in incest. This crime is, of course, his hubris or pride. It is because of his pride that he unwittingly killed his father and slept with his mother. We need to keep in mind that Oedipus did these things unintentionally. If we do, I think there is a good case to say that his crime was hubris.
He is guilty of two crimes, I think.
First, he is guilty of the crime of patricide. That is, he is guilty of the crime of having killed his father.
Second, he is guilty of the crime of incest. He is guilty of this crime because he has married his mother.
In legal terms, he's probably not really guilty of either of these crimes since he had no idea that he had done either of them. However, in the play, he is guilty simply because he has done the deeds, even if he did not realize what he was doing.