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Must is used to convey strong advice imposed by the speaker, or by signs and notice boards.
For the past tense use had to
Mustn't indicates prohibition or strong advice:
Passengers must not smoke on the aircraft
You mustn't work too hard, your health will suffer

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Need to expresses necessity
Don't need to/needn't expresses a lack of necessity
They are similar to don't have to

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Have to is used to refer to strong obligations imposed by another person (figure of authority) rather than the speaker.
Don't have to expresses a lack of obligation, when it is not necessary to do something (the opposite of mustn't, which is prohibition)

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Should/shouldn't + infinitive without to/ ought to

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Should/shouldn't + infinitive without to/ ought to

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Used to express obligation and/or advice. In theory not as strongly as have to/must, but culturally English 'understatement' may mean that this is not the case.
E.g. Your company should upgrade its GMP's ( Good Management Practice) before you submit your product to the FDA (U.S. Food and Drug Administration). Here should = must /have to
You really should see a doctor about that!

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Be supposed to is used to talk about what should be done because of a rule, or because it is expected
E.g. Hey! You're not supposed to have telephones in the exam
Had better + infinitive without to
often warns of possible negative  consequences if the advice/precaution is not taken
E.g. We'd better not eat it-it might be poisonous
You'd better wear a hat, I don't want you to get a cold

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The negative expresses prohibition.
Could/was/were allowed to indicate general permission in the past
In my last job we had flexitime, so we could arrive more or less when we wanted to
Could is not used when referring to a particular situation in the past, only was/were allowed to is possible
E.g I was allowed to stay up late last night

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Let is used to express permission
E.g My dad won't let me watch that programme
Let is not usually used in the passive. Be allowed to is used instead
E.g I wasn't allowed to go to the party on my own
Make is used to express obligation
E.g The teacher made her do some extra homework
In the passive, make is followed by the infinitive with to
E.g He was made to pay for the window he had broken

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Two nouns together describing a single idea. As two words:
E.g post office, hand towel, mouse mat
Or as a single word: postman, handbag, mousetrap
For containers: tea cup, water bottle, ice bucket
For things that occur regularly:
the evening flight, the weekend edition, the January sales