Modals + Perfectives
Perfectives are the (have + past participle form of the verbs). Generally, perfectives are used with the perfect tenses and unreal conditionals. However, they can also be used with modals to indicate past possibility and obligation.
Subject + modal + have + verb in past participle + . . . .
Could/May/Might + Perfective
To indicate a possibility which is already passed you can use these modals with the perfectives.
Subject + could/may/might + have + verb in past participle + . . . .
It might have rained yesterday, but I am not sure.
You could have stayed here last night.
Alex may have reached there by yesterday.
I saw a mobile phone on my sofa; Neel might have left his phone.
We could have gone elsewhere; this place is horrible.
Should + Perfective
To indicate an obligation that was supposed to occur in the past, but for some reason it did not occur. So, it’s the idea of the action being performed in the past but actually it did not occur.
Subject + should + have + verb in past participle + . . . .
You should have come yesterday.
He should not have done this.
I should have completed the task within the due date.
They should not have hit him.
I should have bought that phone.
Must + Perfective
Must + perfective is only used to mean a logical conclusion in the past. It does not indicate any probability or obligation; it indicates a firm conclusion about any past event.
Subject + must + have + verb in past participle + . . . .
It must have rained yesterday.
He must have eaten something unhygienic.
Alex must have gone to sleep.
She must have moved the table.
She must have studied hard.