Iambic Pentameter, for the Most Part
Here's the thing: the writing style of Oedipus the King totally depends upon whose translation you're reading. Since the play is really old, there have been many translations of Sophocles' original Greek through the years. Some of these have kept the play's poetic form, some have translated the thing into prose (non-metrical sentences, instead of poetic lines), and some have even gone for a combo of both.
For the sake of ease, we're sticking with the Francis Storr version, which is why we can say this play definitely has its formal points when it comes to writing. Storr uses a lot of iambic pentameter in his lines:
Explain your mood and purport. Is it dread
Of ill that moves you or a boon ye crave? (I.i.11-12)
Listen to the rhythm of those lines. You might be thinking, "Why did Storr start a new sentence in the first line, then break it off after three words?" Well, iambic pentameter is your answer:
Explain your mood and purport. Is it dread
of ill that moves you or a boon ye crave?
While it may not be a total match with everyday speech patterns, clearly Storr is using this poetic form, which has a kind of regular, formal rhythm to it. Like Shakespeare, the presence of this form often (though not always) indicates that a person of royal or noble character is speaking. Iambic pentameter is reflective of their proper, some might say uptight, demeanor.
BUT, it ain't all da-DUM, da-DUM. When the Chorus comes on the scene, we get lines like:
Hast thou some pain unknown before,
Or with the circling years renewest a penance of yore? (I.i.159-160)
The regular, formal rhythm of the iambic pentameter is dropped, and is instead replaced with rhyming couplets like this one. It's a jolt for the reader, which reminds us that the Chorus is separate from the other characters. Its job is to comment on the behavior that's happening on-stage, and this shift in the writing style, to a less formal, more sing-song-y style, is an effective sign that that's what's going down.