He should've done it.
He didn't do it, but it would have been better if he did.

He had to do it.
He did it, because it was necessary at the time.

He must've done it.
He probably did it.

He shouldn't have done it.
He did it, but it would have been better if he didn't.

He didn't have to do it.
He may or may not did it, but it was not necessary at the time.

He mustn't've done it.
He probably didn't do it.