can/can't
could/couldn't
may/may not
shall/shall not
will/won't
have to/don't have to
must/must not
should/shouldn't

ought to/ought not to
might/might not
used to/didn't use to
would/wouldn't

prediction of future events

He'll (= will) forget his umbrella if you don't remind him.
What will it be like, living in the 21st century?
We'll (=will/shall) all be dead in a hundred years.
Stop crying! It won't make things any better, you know.

Shall can be used with first person singular (I) and first person plural (we). However, it is less common than will, especially in American English.

personal intention

I'll (=will/shall) be back in a minute.
I won't/shan't ever speak to him again.
We will/shall overcome all difficulties.

Shall can be used with I and we, but is less common than will, especially in American English.

willingness, wish

Will/would you help me with my homework? (request)
No. I won't. (refusal)
I'll (=will) do it for you if you like. (offer)
Shall I give you a hand with the dishes? (BrE) (offer)
Shall we buy her a present? (suggestion)

In British English, first person questions expressing willingness or wish use shall (Shall I/we? = Do you wish me/us to...?) First person statements use will (I/we will)

Note that shall is not usually used in this way in American English.

ability

I can speak Chinese, but I can't write it.
She could swim for miles when she was younger.
Can/Could you close the window, please? (request)

Could is used to talk about ability, NOT about particular events which actually happened in the past. Verbs like manage to are used instead. She finally managed to pass the exam.

Polite requests are often made by appearing to ask about ability with can and could.

permission

Can/May I have another piece of cake, Dad? (request)
No you can't. You'll make yourself sick.
Do you think I could leave early tonight? (request)
You can/may leave at 5:30 if you like.
I'm afraid you can't leave until you've finished that work.
Might I have a word with you? (BrE) (formal request)

Can is commonly used to ask for or give permission. May is more formal.

Could and might are used to ask for (not to give) permission. They are more tentative than can.

unreality, hypothesis

I would love to travel round the world. (if I had the chance)
What would you do if you won a lot of money?
I wouldn't have gone, if I'd known he was going to be there.
Would you like some tea (if I made some)? (invitation)
Should he protest (if he protested), what would you say? (fml)

Would is commonly used in the main clause of conditional sentences to show that a situation is unreal or tentative.

Because it can express tentativeness, would is also used in polite invitations, offers, and requests.

possibility

She may/might (not) go to Paris tomorrow
They may/might (not) be meeting her.
Joe may have/might have missed the train.
Where can/could they be?
You can't have forgotten my birthday!
Learning English can be fun. (= is sometimes fun)
Don't touch that wire. It could be dangerous.
They could have had an accident, I suppose.

Could suggests that something is less likely than may or might.

When it expresses possibility, can is most often used in question forms: What can have happened? However it is also used to express general possibility in sentences where its meaning is similar to "sometimes": His behavior can make us laugh. (= sometimes makes us laugh)

Can't and can't have are used to show that there is no possibility. (See certainty below)

certainty

Joe must be at least 45. I'm sure he's at least 45.
No, he can't be over 40. I'm sure he isn't over 40.
He must have graduated years ago. I'm sure he graduated years ago.
We can't have been at college together. I'm sure we weren't at college together.

Must have is the past form of must when it is used to express certainty.

Must and must have express stronger certainty than will and will have.

Can't and can't have express stronger certainty than won't and won't have.

obligation, requirement

You must finish this job by tomorrow.
I must phone my parents tonight.
He had to finish the job by the next day.
You don't have to/don't need to/needn't (BrE) do it until next week. (= it is not necessary)
You must not smoke in the cinema. (= it is forbidden)
I didn't need to/didn't have to get up early this morning. (= a the speaker did not get up early, or b the speaker did, in fact, get up early)
You needn't have bought me a present. (BrE) (= but you did buy a present)

Had to is the past form of must when it is used to express obligation.

Don't have to/don't need to/needn't (BrE) are used to show that there is no obligation. Must not is used to show that there is an obligation not to do something.

The contracted forms needn't and mustn't are common in British English but rarely used in American English.

desirability

You should/ought to give up smoking. (advice)
We should/ought to go to that new Japanese restaurant sometime. (suggestion)
The farmers should have/ought to have been consulted. (but they were not consulted)
You shouldn't/ought not to work so hard, you know.

The contracted form oughtn't is common in British English but rarely used in American English.

probability

Their meeting should/ought to be over now. (= I expect it is)
He should/ought to be home at 5 o'clock today. (= I expect he will be)
They should have/ought to have received our letter by now. (= I expect they have)

In this meaning should and ought to are not as strong as will and must (see certainty above)