Should 2

We can use 'should' after 'reporting verbs' such as
demand  insist  propose  recommend  suggest

• He demanded that we should pay for the repair.
• She insisted that she should pay for the meal.
• I have proposed that he should take charge of the organization.
• The committee recommends that Jane should be appointed.
• We have suggested that Michael should be given a reward for his hard work.

However, it is also possible to say exactly the same thing by omitting the 'should' and just using the infinitive form without 'to' . Some people call this the 'subjunctive' form.

• He demanded that we pay for the repair.
• She insisted that she pay for the meal.
• I have proposed that he take charge of the organization.
• The committee recommends that Jane be appointed.
• We have suggested that Michael be given a reward for his hard work.

We can use 'should' after various adjectives. Typical examples are :
funny  interesting  natural  odd  strange  surprised  surprising  typical

• It's funny that you should say that. I was thinking exactly the same thing.
• It's interesting that they should offer him the job. Not an obvious choice.
• It's natural that you should be anxious. Nobody likes speaking in public.
• Isn't it odd that he should be going to the same tiny hotel? What a coincidence.
• It's strange that you should think so. Nobody else does.

We can use 'should' in 'if clauses' when we believe that the possibility of something happening is small.

• If you should happen to see him before I do, can you tell him that I want to speak to him urgently?
• If there should be a problem, just give me a call and I'll sort it out.
• If anyone should ask where I am, say I'm in a meeting.

We use 'should' in various fixed expressions.
To show strong agreement

• They're paying you compensation?   I should think so.

To express pleasure when you receive a gift

• What a fantastic present. You really shouldn't have.

To emphasize  a visible emotion

• You should have seen the look on her face when she found out that she had got the promotion.