Form and rules
How it works
Must/have to can express necessity; must often sounds speaker-imposed or strongly stated, while have to readily describes external rules and works across tenses.
Mustn't means prohibition; don't have to/needn't means lack of necessity, not prohibition.
Should/ought to express advice or weaker obligation; be supposed to expresses expectation, arrangement or rule.
Permission can use can, could or may depending on context/formality; be allowed to is useful across tenses.
Past/future meanings often require had to, was allowed to, will have to or will be allowed to rather than a simple modal form.