Auxiliary

Uses

Present / future

Past

may

1. Polite request

May I borrow your pen?

2. Formal permission

You may leave the room.

3. Less than 50% certainty

A: Where's Tom?
B: He may be at the office.

He may have been at the office.

might

1. Less than 50% certainty

A: Where's Tom?
B: He might be at the office.

He might have been at the office.

2. polite request

Might I borrow your pen?

should

1. advisability

I should help my father in the morning.

I should have helped my father yesterday.

2. 90% certainty

She should pass the exam! (future only, not present)

She should have passed the exam!

ought to

1. advisability

I ought to study tonight

I ought to have studied last night

2. 90% certainty

She ought to do well on the test. (future only, not present)

She ought to have studied last night.

had better

1. advisability with threat of bad result

You had better be on time, or we will leave without you.

(past form uncomon)

be supposed to

1. expectation

He is supposed to come at 12.

He was supposed to come at 12.

be to

1. strong expectation

You are to be here at 9:00.

You were to be here at 9:00

must

1. strong necessity

I must go to class today

I had to go to class yesterday

2. prohibition (negative)

You must not open that door.

3. 95% certainty

Mary isn't in class. She must be sick (present only)

Mary must have been sick yesterday.

have to

1. necessity

I have to go to class today.

I had to go to class yesterday

2. lack of necessity (negative)

I don't have to go to class today.

I had to go to class yesterday.

have got to 

1. necessity

I have got to go to class today.

I had to go to class yesterday.

will

1. 100% certainty

He will be here at 6:00 ((future only)

2. willingness

The phone's ringing.
I'll get it.

3. polite request

Will you please pass the salt?

be going to

1. 100% certainty

He is going to be here at 6:00 (future only)

2. definite plan

I'm going to paint my bedroom. (future only)

I was going to paint my room, but I didn't have time.

can

1. ability / possibility

I can run fast.

I could run fast when I was a child but now I can't.

2. informal permission

You can use my car tomorrow.

3. informal polite request

Can I borrow your pen?

4. impossibility (negative only)

That can't be true!

That can't have been true!

could

1. past ability

I could run fast when I was a child.

2. polite request

Could I borrow your pen?
Could you help me?

3. suggestion

A: I need help in English.
B: You could talk to your English teacher.

You could have talked to your teacher.

4. less than 50% certainty

A: Where's John?
B: He could be at home.

He could have been at home.

5. impossibility (negative only).

That couldn't be true!

That couldn't have been true!

be able to 

1. ability

I am able to help you. I will be able to help you.

I was able to help him.

would

1. polite request

Would you please ass salt? Would you mind if I left early?

2. preference

I would rather go to the park than stay home.

I would rather have gone to the park.

3. repeated action in the past

When I was a child, I would visit my grandmother every weekend.

used to

1. repeated action in the past

When I was a child, I would visit my grandmother every weekend.

shall

1. polite question to make a suggestions

Shall I open the window?

2. future with "I" or "we" as subject

I shall leave at nine. (will = more common)