What are some examples where peripeteia and anagnorisis occur in the play Oedipus Rex?

According to Aristotle in Poetics, a tragedy depicts the downfall of a noble tragic hero, which occurs through a combination of the hero's tragic flaw (usually hubris), fate, and the will of the gods.
The tragic hero is driven to accomplish a goal (for Oedipus, this goal is to find the person who killed Laius) but inevitably encounters limits to achieving that goal. These limits can come in the form of their tragic flaw or mistakes (hamartia), failure of reason or the hero's own hubris, the will of the gods expressed through oracles or prophets, or nature itself.
The tragic hero doesn't necessarily die at the end of the play (Oedipus doesn't die at the end of Oedipus Rex), but they must undergo peripeteia, which is a significant change in their life or an unexpected reversal of fortune.
According to Aristotle, the reversal of fortune, the peripeteia, "should arise from the internal structure of the plot, so that what follows should be the necessary or probable result of the preceding action."
Aristotle himself tells us when the peripeteia occurs in Oedipus Rex.

Thus in Oedipus, the messenger comes to cheer Oedipus and free him from his alarms about his mother, but by revealing who he is, he produces the opposite effect.

During the course of the play the hero also experiences anagnorisis, which is a startling discovery or revelation that causes the hero to undergo a change from ignorance to knowledge. This discovery can be a revelation made by other characters to the hero or the hero's own discovery about fate and the will of the gods.
Anagnorisis is the moment in the play when the hero recognizes their true nature or identity, realizes identity of other characters in the play, or discovers the reality of their situation or that of other characters, which ultimately leads to the resolution of the play.

But, of all recognitions, the best is that which arises from the incidents themselves, where the startling discovery is made by natural means. Such is that in the Oedipus of Sophocles . . .

In Oedipus Rex, anagnorisis occurs naturally, "from the incidents themselves," which is to say without contrived plot reversals or deus ex machina.
What is significant about Oedipus Rex, and what makes it so effective as a tragedy, is that the peripeteia and the anagnorisis occur at the same time.
Everything about Oedipus, Laius, Jocasta, Polybus, and Merope is revealed as a result of the Messenger's information about Polybus's death and the Herdsman's confirmation that Oedipus was the son of Laius and Jocasta. These revelations immediately precede Oedipus's realization that he did, in fact, kill his father and marry his mother.
For Oedipus, his downfall is a result of his unwittingly fulfilling a prophecy in trying to escape it.