Which modal and when ?

 I) Abilities

You can also use: 'be able to'.

=>CAN:

=>CANNOT// CAN'T

=>COULD (in the past)

=> COULD (hypothesis)

=> hypothesis in the past

COULD + HAVE + past participle

She can swim.

I can't / cannot speak Spanish.

He could run 15 kilometres.

You could succeed if you worked harder.

Your children could have helped you.

II) Being allowed to do something:

CAN:

MAY:

CANNOT/ MAY NOT/ MUSTN'T:

TO BE ALLOWED TO:

Can I borrow your pen?

May I borrow your pen?

You cannot// mustn't walk on the grass.

 
You will be allowed to use your copybook to take your test.

III) Certainty:

MUST:

=> present simple :

=> present continuous:

=> in the past:

You must be tired.

He must be sleeping.

She must have been very upset.

IV) I'm also sure it is not true:

CAN’T

COULDN’T

It can't be true!

He couldn't have missed his train.

V) It can happen (50% / 50%) :

MAY/ MIGHT

=> present 

=> present be+ing

=> in the past

COULD

He may be in his room.

She may be working.

They might have met this morning.

VI) Obligations:

- MUST

- HAVE TO

=>after a modal  

=>with several tenses

I must do my homework!

 I have to stop smoking my doctor said.

I have to work or I might be late.

You will have to work.

I had to lie down.

I've had to lie down. 

VII) Advice:

SHOULD

 You should do your homework.

You should have phoned her.

VIII) Conditional:

WOULD

=> conditional

=> past conditional

=> with an if-clause:

=> being polite

You would be pleased to meet him.

You would have been pleased to meet him.

If I were you, I would go to sleep.

Would you like some tea?

IX) Other modals:

NEED

SHALL

You needn't say that!!  

He needn't have told me!!

Shall we go to the cinema?

You shall not be rude!