Is it "What should he have done?" or "had he done"?

• What should he have done?
• What should he had done?

Could you tell me which one is correct? (If any.)

What you want is

What should he have done?

Keep in mind that your basic present indicative form is He should [VERB], where the lexical VERB is in the infinitive, uninflected form. For instance:

He should throw the ball.
He should work harder.
He should do this.

When you make a question of this, you put your interrogative at the front, keep the modal verb should in second place, and move the Actor, he, after the modal verb:

What should he throw?
How should he work?
What should he do?

Now—when you cast your basic form into the past, you employ the present perfect form, using the infinitive of the auxiliary HAVE plus the past participle of the lexical verb:

He should have thrown the ball.
He should have worked harder.
He should have done this.

And the process of turning that into a question is the same as before—put your interrogative at the front, keep the modal verb should in second place, and move the Actor, he, after the modal verb:

What should he have thrown?
How should he have worked?
What should he have done?

You can only say the first. That is because a modal verb such as should is followed by the plain form of the verb such as have, and not by an inflected form such as had.