Something I feel should be part of Lex Luthor's characterization that I don't think is ever explored enough:
He has no interest in discovering Superman's secret identity.
He does not believe that Superman HAS a secret identity.
Why would he? Superman gives lengthy, tell-all interviews to the Daily Planet in which he makes it clear that his name is Kal-El, that he is an alien from the planet Krypton, and that he stays in a crystal fortress in the Antarctic.
Lex has no reason to suspect that Superman might even desire a normal human life, with a wife and son and office job in a dying medium. He's an alien from another planet with immeasurable abilities and strength. All that wholesome aw-shucks posturing is just an act; obviously Superman would never want to live among humans, he wants to dominate them.
After all, if Lex had Superman's power, that's exactly what he would do with it.
Lex is a narcissist! He can barely see beyond his own nose, and he would never, ever, for one moment suspect that Superman would actually be Clark Kent, that straightlaced, soft-spoken journalist who still wears double-breasted suits. Everyone knows his wife wears the pants in that relationship.
Even the fact that Superman is mostly seen in Metropolis wouldn't seem suspicious to Lex. After all, in his mind, he's the only person who has Superman's number, the single biggest threat to his unchallenged superiority over mankind - and he lives in Metropolis too.
