must – imperative – something that is necessary and personal. It has no other tenses.

I must see a doctor. I feel very ill.

I must finish my project today. (I personally consider that I must finish the project today)

Must not = mustn’t = it is totally forbidden

You mustn’t use the mobile phone when you are driving. (it is formal)

In spoken English, people say you can’t use your mobile phone when you’re driving.

should – it is used to express a piece of advice or personal opinion. It has no other tenses.

You should go home now, it’s late. (this is my personal opinion or a piece of advice)

You should finish the project this week. (it is my personal opinion)

should not = shouldn’t = it is used to express a piece of advice or personal opinion

You shouldn’t miss this class. We do the review. (in my opinion, it is important to you to be present at the review)

have to – less strong than must, mainly expresses general obligations. It is used more in spoken English. Many times it is used as a synonym of ’must’. ’Have to’ expresses that the situation asks me to do something, it is more situational than personal.

I have to pick up a package from the post office.

not have to – expresses that it isn’t necessary to do something, but it is not forbidden.

You don’t have to be here at 7 o’clock. (but it is not forbidden)

You don’t have to wash the dishes, I’ll do it later.

It can be used in different tenses: past, future.

need to – it is necessary because there is a need. Many times it is synonym with ’must’ or ’have to’. It is semi-modal, because of ’to’.

She needs to lose some weight till the wedding.

I need to buy something to eat, the fridge is empty.

I need to buy larger clothes.

It can be used in future.

ought to – it expresses a duty to do something. It can be synonym with ’need to’ and ’should’ but it doesn’t express the speaker’s opinion.

You ought to take care of your grandmother. She has nobody else than you.

You ought to attend that meeting. (it is not a piece of advice, it is expected from you to be there)

to be supposed to – it expresses the fact that somebody is in charge of something

I am supposed to make some changes to this project. (it is not my will, it is my task)

had better = ’d better – it expresses a warning, not a piece of advice, not a personal opinion.

You’d better finish this project today! (it is a warning)