The most predominant conflict in Oedipus Rex is character vs. fate. Oedipus was born to a cursed family and was therefore cursed himself. According to myth, King Laius was King Pelops' charge. Laius became such close friends with Pelops' youngest son Chrisippus that they ran away together. As a result, Pelops cursed Laius (Tripp, Meridian Handbook of Classical Mythology). Hence, it is as a result of Pelops' curse that Oedipus is born to the fate of killing his own father, Laius, and sleeping with his own mother. Therefore, the most predominant conflict reflects circumstances far beyond Oedipus's control, making the predominant conflict character vs. fate.
However, this would not be a true Greek tragedy, particularly not the tragedy Aristotle lauds in his Poetics as the most perfect tragedy, if Oedipus Rex did not also possess a character flaw that helps propel him towards his great fall (Barbara F. McManus, "Outline of Aristotle's Theory"). Oedipus's character flaw is recognized to be his excessive pride, making the second predominant conflict in the play character vs. self. We particularly see evidence of Oedipus's pride in his dealings with both Creon and Tiresias. It is Oedipus's pride that drives him to suspect a plot to overthrow, not only Laius, but himself and to disbelieve Tiresias's prophecy. Instead, Oedipus believes that both Creon and Tiresias are conspiring against him and that Creon paid Tiresias to deliver false prophecy, as we see in Oedipus's lines:
... the trusted Creon, my friend from the beginning, beguiles me and secretly desires to oust me, engaging this craftily-working wizard ... who sees clearly only for profit. (405-409)
It is also Oedipus's pride and even his impetuous temper that leads him to kill a man who ran him off the road at the intersection heading towards Delphi--a man that turned out to be his own father. Therefore, the second most predominant conflict in the play is character vs. self.
The major conflict in this play is between reality and the desperate attempts of Oedipus and some of the other characters, most notably Jocasta, to ignore the facts that point to one inevitable conclusion: that Oedipus is the man responsible for the death of the former king of Thebes, and therefore the man responsible for the plague that is troubling Thebes. Note what Oedipus says in the following quote when his suspicions about his own identity increase massively:
Oh, but if there is any blood-tie
between Laius and this stranger...
what man alive more miserable than I?
More hated by the gods?
Of course, the play gradually presents both the audience and Oedipus with evidence that becomes more and more unavoidable, pointing the finger directly at Oedipus and forcing him finally to face the facts about his own identity, with tragic results. Reality wins out. The play thus places the forces of reality against the forces of intentional ignorance, or trying to ignore the facts for as long as possible. Both Jocasta and Oedipus finally have to accept the truth, and this brings disaster into their lives.