Grammar - Should

Should
Should is an auxiliary verb - a modal auxiliary verb.

1. To give advice, a recommendation or a suggestion
You should see the new movie. It's great!
You should stop eating too much.
What should I wear?

2. Expresses an obligation that is not as strong as Must.
You should be wearing your seat belt.
I should be at work now. (duty)
You should pay more attention in class.

3.Talk about probability and expectation
Are you ready? The bus should be here soon.
He should have the letter by now. I sent it a weeks ago.
It should be fine tomorrow.

4.To ask questions
Should we turn left at this street?

5.Should for conditionals
We sometimes use should (instead of would) for the first person singular and plural (I, we) of some conditionals:
If I lost my job I should have no money.

Shouldn't
We shouldn't leave without saying goodbye.
Are you tired? You shouldn't work so much.

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Grammar - Could

Could

Could is an auxiliary verb, a modal auxiliary verb.
Could is similar to Can and often replaces Can in the past tense (though not always).

To express ability in the past
I could ride a horse when I was nine but now I can't.
He could write when he was three years old.

Polite word used to ask for permission or to request something (in the present)
Could you pass me the salt please?
Could I borrow your car?
Could you call back later?

General permission in the past
He couldn't go to the stadium because his father died.
We couldn't enter our company without employee ID card.

A suggestion when asked what to do (choices and opportunities)
He could try and fix it himself.
We could watch live footbal match on TV if you are interested.

Couldn't: Sure that something is untrue
Just like can't, you can use couldn't when you are sure that something isn't true or real.
Where are they? They could have got lost.
It's ten o'clock. They could have arrived by now.

Could + have + past participle: To express a possibility in the past
Why did you do that? You could have lost your eye.

Could + have + past participle: Unrealised past ability
She could have married anyone she wanted to.

Grammar - Can

Can
Can is an auxiliary verb, a modal auxiliary verb.

Can expresses ability. Cannot (can’t) shows inability.
She can speak four languages.
She can't dance very well.
Birds can fly.

To express a possibility (in general) - not for future possibilities. For this you would use may or might.
It can get cold there at night so take your coat.
I know you can win beat him.

To offer to do something for others
Can I help you?
Can I do that for him?

To ask for or give permission / To request something
You can borrow my bag, I don't need it right now.
Can I ask you a question?
You can't go to the zoo. It will still be wet from the rain last night.

Could is more formal and polite than can:
Could I ask a question please?
Could we go to school now?

Can't: Something that is forbidden or not allowed
You can't smoke in the school.
You cannot drive a car without a license.

Can't: when you are sure that something isn't true or something is surprising
It can't be cold outside! It's 7th of August!
They can't have landed on the March.

The negative form is can't in spoken English and cannot in written English.