Should in subordinate clauses
Should is often used in that-clauses after adjectives and nouns expressing the importance of an action. Adjectives that can be used like this are: important, necessary, vital, essential, eager, anxious, concerned, wish etc.
It is important that she should behave.
It is important that she should learn the importance of being punctual.
It is essential that she should talk to the manager.
It is necessary that she should be informed.
It is important that the meeting should start on time.
I am anxious that nobody should get hurt.
Verbs expressing similar meanings can also be followed by that-clauses with should.
She insisted that she should be paid at once.
She suggested that I should consult a doctor.
He recommended that we should increase our productivity.
Should is also used in subordinate clauses after words expressing personal judgments and reactions.
It is astonishing that she should behave so rudely.
I was shocked that she should say that sort of thing to her mother.
In American English, would is more common in this case.
It is astonishing that she would behave so rudely.
Should can also be used in if-clauses, after in case, so that, in order that and lest.
If you should see Kathy, give her my love.
He turned the music down so that he shouldn’t disturb his neighbors.
I have bought a bottle of wine in case Peter should come.
This use of should, too, is more common in British English than in American English.