In indirect speech

Should is the past tense of shall in indirect speech.

Direct speech: The Captain said, “Players shall assemble at the ground at 4 pm.”
Indirect speech: The Captain said that players should assemble at the ground at 4 pm.
Direct Speech: He said, “The scoundrel shall be punished.”
Indirect speech: He said that the scoundrel should be punished.

To express duty or obligation

Should is used with pronouns of all the three persons to express duty and obligation.

We should respect our parents and teachers.
We should help them.
I should go now.
You should not say such things to her.
He should be punctual.
You should pay the fees in time.

In conditional clauses

Should can be used in conditional clauses expressing possibilities and suppositions.

If she should come, ask her to wait.
Should it rain, we will not go out.

Note that the clause ‘If she should come’ expresses less likelihood of her coming than the clause ‘if she comes’. And hence the sentence means that there is not much chance of her coming. But if she turns up ask her to wait.

Should expresses less possibility than shall.

I shall be happy to help you.
I should be happy to help you.

Here the first sentence expresses a greater possibility of my helping you than the second sentence.

Should is often used in result clauses which are preceded or followed by a conditional clause expressing an unreal situation.

If I were you, I should accept this offer. (Unreal because I am not you.)
No John, I shouldn’t quit if I were you.

After lest

Should is the only auxiliary that may be used after lest.

The police surrounded the building lest the thieves should escape.

Work hard lest you should fail.

Should is also used in the expression ‘should like to’ which is a polite form of making a statement.

I should like to thank my colleagues for giving me this opportunity to…
I should just like to mention that we need to act fast.

To express possibility or likelihood

Should can be used to express possibility or likelihood.

I should be able to win.
She should be coming.

Should have + past participle

The structure should have + past participle is used to talk about past events that did not happen. It is also used to criticize people for not doing things.

She should have asked my permission. (She didn’t ask my permission.)
You should have helped them. (You didn’t help them, but you should have.)