SHOULD

Should + perfectThe aspect expressing completion. E.g.: He’s finished university. infinitive can be used to emphasisSpecial importance or attention given to something. E.g.: All I wanna do is have some fun.e unlikelihood in the past. This use is rather formalNot belonging to colloquial or everyday speech; official. E.g.: I demand that Mr Smith call the person on whom the committee has decided. or may even sound archaic to some speakers.

would rather/sooner + perfect infinitive

This structureThe way in which the parts of a sentence, clause or expression are arranged. E.g.: make somebody + infinitive in "The teacher made me rewrite the composition." can express preferences in the past.

Would you rather have had pork than veal?
I’d rather not have called the police.

If we delete the word rather, we get the main clauseA clause that could be a sentence in itself. E.g.: I wouldn’t go there if I were you. ("I wouldn't go there") When I arrived he was surfing. ("he was surfing") of a third conditional sentence.

Should + perfectThe aspect expressing completion. E.g.: He’s finished university. infinitive is used as a polite expressionA word or group of words used in a particular situation. E.g.: Good morning, at six, on the sofa of thanks on receiving a gift or a favour.

"I've brought some wine."
"Oh, you really shouldn't have!" 

This is a special use of could, which expresses possibility.

He's an angel! He couldn't be more polite.
We would drink a cup of fresh cow milk and lie in the tall grass every morning; we couldn't have been happier...

Would can express insistence/willingness and wouldn't refusal/unwillingness in the past.

I had asked Joe to leave me more space but he would park his car next to mine, so I couldn't get out.
Try as I might, the door wouldn't open.

Couldn't + perfectThe aspect expressing completion. E.g.: He’s finished university. infinitive can also express unwillingness.

To express past expectation and assumption, the same auxiliariesA helping verb used with main verbs to express mood, voice and tense. E.g.: I can't stand when I am disturbed. ("can", "am") are used in the case of present expectations and assumptions. However, the infinitives that follow are perfectThe aspect expressing completion. E.g.: He’s finished university. infinitives.

Compare these two sentences:

I could have given you a lift.

This sentence is an offer only in theory. In practice it expresses possibility. Note that this is the main clauseA clause that could be a sentence in itself. E.g.: I wouldn’t go there if I were you. ("I wouldn't go there") When I arrived he was surfing. ("he was surfing") of a third conditional sentence. 

This structureThe way in which the parts of a sentence, clause or expression are arranged. E.g.: make somebody + infinitive in "The teacher made me rewrite the composition." can express arranged eventA general term expressing that somebody or something does something or that something happens. E.g.: An apple fell on Isaac’s head.s.

The concert was to begin at 6.

However, if the planned event did not happen, we use be to + perfectThe aspect expressing completion. E.g.: He’s finished university. infinitive:

You were to have tidied your room this morning.

You should have been on time. (but you were not)
We ought to have protected the environment. (but we did not)

These can also be considered as criticism or advice.