Past Modals

We use past modals to express a possibility in the past and to give advice or suggestion to a
past situation.

The structure is as follows:
Subject + Modal Verb + Have + Verb in Past Participle + Object

Peter went to the party. (100% sure)

Peter must have gone to the party because he wanted to go and he isn’t at home now. 
(95% possibility)

In the first example, the speaker is 100% certain that Peter is at the party even though no reason was given. In the second example, the speaker is not completely certain, but they have a very strong reason to believe what they are saying.

The launch of the rocket was called off because something happened to the engine. (100% sure)

Something could have happened to the engine of the rocket because they called the launch off. 
(50% possibility)

Something might have happened to the engine of the rocket because they called the launch off. 
(50% possibility)

Something may have happened to the engine of the rocket because they called the launch off. 
(50 % possibility)
In the first example, the speaker is absolutely certain why the rocket launch was called off. Maybe there was confirmation by scientists or flight engineers. In the second example, the speaker doesn’t have enough information of why the launch was called off, they are only speculating.

Ralph wasn’t at the meeting because he called and said he had problems with his car. (100% sure)

Ralph wasn’t at the meeting because he could have had problems with his car. 
(50 % possibility)
Ralph wasn’t at the meeting because he might have had problems with his car. 
(50 % possibility)

Ralph wasn’t at the meeting because he may have had problems with his car. 
(50 % possibility)

In the first example, the speaker is certain of what happened to Ralph because they spoke to Ralph directly. In the following example, they are only guessing what happened to Ralph.
Examples of suggestions or advice regarding a past situation. In this case we only use the modal verbs “should”, “ought to” and “could”.

The structure is as follows:

Subject + Modal Verb + Have + Verb in Past Participle + Object

Situation 1: Your friend took his father’s car without permission. He got into an accident with the car.

You might say to your friend:

You shouldn’t have taken your father’s car without his permission. 

Situation 2: My friends wanted to go to the beach, but the roads were a mess from the heavy rains. Now they are sad because they are having a boring weekend.

You might say to your friends:

You guys could have gone to the mountains instead. 
Situation 3: My friend’s briefcase was in his/her car. Now it has disappeared.

You might say to your friend:

You ought to have taken it with you. 

With the modal verb “ought to” is always with the particle “to”