Modality: must, may, should, be able

Two shapes

Some modal words conjugate and take a nominative subject; most do not, and put the person in the dative. Getting the shape right matters more than the meaning nuance.

Word	Means	Shape
магу, можаш...	can, be able	personal: Я магу прыйсці.
мушу, мусіш...	must (strong)	personal: Я мушу ісці.
умею	know how to	personal: Я ўмею плаваць.
трэба	need to, must	impersonal: Мне трэба ісці.
можна	may, be allowed	impersonal: Тут можна курыць?
нельга	may not, cannot	impersonal: Тут нельга.
варта	be worth	impersonal: Варта паглядзець.
мае / маю	be due to	personal: Ён мае прыйсці а пятай.

Мушу and мае

Two that a Russian speaker will not predict:

Мусіць is the strong obligation verb - stronger than трэба, closer to have to with no way out: Мушу прызнацца - I have to admit. As a parenthetical, мусіць also means probably: Ён, мусіць, ужо дома.
Мець plus an infinitive expresses a scheduled or expected action: Цягнік мае прыйсці а трэцяй - the train is due at three. This is a Western Slavic pattern that Belarusian shares with Polish and Russian does not have.

Tense on impersonals

The impersonal ones take было and будзе for past and future, and the person stays in the dative throughout:

Мне трэба было ісці. - I had to go.
Табе можна будзе адпачыць. - You will be allowed to rest.

Softening

Adding бы makes any of them tentative: Варта было б спытаць, Трэба было б патэлефанаваць. This is the ordinary way to give advice without pressing.

Frequently asked questions

When is Modality: must, may, should, be able used?

Some modal words conjugate and take a nominative subject; most do not, and put the person in the dative. Getting the shape right matters more than the meaning nuance.

How is Modality: must, may, should, be able formed?

The impersonal ones take было and будзе for past and future, and the person stays in the dative throughout: Adding бы makes any of them tentative: Варта было б спытаць , Трэба было б патэлефанаваць . This is the ordinary way to give advice without pressing.

What are common mistakes with Modality: must, may, should, be able?

Typical errors: making the person the nominative subject of трэба or можна, and missing that мець plus infinitive means scheduled rather than possessed.

Sources

Кароткая граматыка беларускай мовы. У 2 ч. Ч. 1: Фаналогія. Марфаналогія. Марфалогія / навук. рэд. А. А. Лукашанец — НАН Беларусі, Інстытут мовазнаўства імя Якуба Коласа; Мінск: Беларуская навука, 351 с., 2007

The current academic reference grammar; the linked edition serves the full text. Its morphology part is the source for the paradigms used here. On the neuter noun it states the nominative singular endings as -а (-о) and -е (-ё), and gives the nominative plural as -ы after a hard consonant (сёлы, вёдры) against -і after a soft consonant and after к, х (палі, воблакі).