Modals in the Past

can't have

certainty

e.g. I can't have lost my keys! (I'm sure I didn't)

could	permission e.g. When I was a teenager, I could go out as late as I wanted.
	general ability e.g. Tom could walk by the age of 8 months.

could have	possibility, but did not happen e.g. I could have passed my driving test if I'd really tried.
	uncertainty e.g. I guess it could have been Sandra on the phone.

couldn't	permission e.g. When I was a teenager, I couldn't stay out as late as I wanted.
	general ability e.g. I couldn't walk until I was 2.
	ability in a particular situation e.g. I tried hard but I couldn't persuade him to go to the party with us.

couldn't have	possibility/ability e.g. I couldn't possibly have passed my driving test, even if I'd tried harder.
	uncertainty e.g. It couldn't have been Sandra on the phone, could it?
	with comparative adjectives e.g. I couldn't have asked for better weather on my wedding day.
	unwillingness e.g. I couldn't have left the dog in the car for long (so I didn't).

didn’t need to

unnecessary action not done

e.g. As I was alone this weekend, I didn't need to do any cooking (so I didn't).

had to

obligation (past form of must)

e.g. I couldn't go out last night because I had to do my homework.

may have

uncertainty

e.g. I guess I may have been a bit hard on her when she came home an hour late.

may not have

uncertainty

e.g. He may not have found out yet that he has passed the test.

might have	possibility (didn't happen) e.g. You might have been killed!
	uncertainty e.g. I guess I might have been a bit hard on her when she came home an hour late.
	annoyance at someone's failure to do something e.g. You might have told me that you had invited all your colleagues round for dinner!
	might have known + would (idiom to ironically express that somebody's action was typical) e.g. I might have known that he would finish with me as soon as he found out I wasn't wealthy!

might not have

uncertainty

e.g. He may not have found out yet that he has passed the test.

must have	certainty e.g. He must have known how much it was going to cost. (I'm sure he knew.)
	with surely in exclamations e.g. Surely he must have known how much it was going to cost!

needn't have

unnecessary action that was actually done

e.g. Oh, you needn't have done the washing up!

ought not to have

criticism (more common is shouldn't have)

e.g. You ought not to have been so frank with him.

ought to have

expectation (should have is more common)

Why is she late? She ought to have arrived by now!

should have	expectation Why is she late? She should have arrived by now!
	should have + verbs of thinking e.g. I should have thought you knew.
	with be and adjectives, describing chance e.g. It was weird that you should have been staying in the same hostel last month.
	criticism (you didn't do something, but it would have been the right thing to do) e.g. The party was such fun last night! You should have come!

shouldn't have

criticism

e.g. You shouldn't have been so frank with him.

polite expression of thanks on receiving a gift or favour

"Here's a bottle of wine for you"

"Oh, you really shouldn't have!"

would have	events in the past that did not happen e.g. I wouldn't have gone out with him, but he didn't ask me.
	assumptions e.g. Oh, that would have been Sarah on the phone just now.