Modals (can/could/may/might/must/should)

Structure

Modal + base verb (V1) — no "to", no "s" for third person

Ability: "I can swim." (present) / "I could swim as a child." (past ability)
Permission (informal): "Can I leave early?"
Polite request: "Could you help me?" (more polite than "can")
Possibility: "It could rain later."

Permission (formal): "May I come in?"
Possibility (may = more likely, might = less certain/more hypothetical): "It may rain." / "It might rain, but I doubt it."
Wishes (formal): "May you live long."

Obligation from speaker's own authority: "You must finish this today." (I insist)
Obligation from external rule: "You have to wear a uniform." (school rule, not speaker's choice)
Strong logical deduction (certainty): "He must be tired after that journey."
Negative forms differ in meaning:
- "You mustn't smoke here." (prohibition — not allowed)
- "You don't have to come." (no obligation — optional)

Advice/recommendation: "You should see a doctor."
Expectation: "The package should arrive tomorrow."
Past regret/criticism (should have + V3): "You should have told me earlier!" (but you didn't)
"Ought to" = same meaning as "should", slightly more formal

No obligation: "You needn't come if you're busy."
needn't have + V3 = action was done but was unnecessary: "You needn't have brought a gift." (you brought one, but it wasn't needed)
didn't need to + V1 = action was not necessary (may or may not have happened): "I didn't need to wake up early." (ambiguous whether I woke up early or not)

Polite request: "Would you mind closing the door?"
Past habitual action: "He would often visit us on Sundays." (habit, similar to "used to")
Conditional (2nd/3rd conditional): "If I had money, I would travel."

must have + V3 — certainty about the past: "He must have left already."
might/may have + V3 — possibility about the past: "She might have missed the bus."
should have + V3 — regret/criticism about a past action not done: "You should have studied harder."
could have + V3 — past ability that wasn't used: "I could have won, but I gave up."
needn't have + V3 — unnecessary action that was done (see above)

Modal + be + V3

You ___ finish this by tomorrow. (must/may) — must
___ I come in? (Can/Should) — Can/May
She ___ be at home now; her lights are on. (must/might) — must
He ___ have missed the bus — he's not usually late. (might/should) — might
You ___ have told me earlier! (should/can) — should
Students ___ not use phones during exams. (must/could) — must
It ___ rain later, take an umbrella. (may/must) — may
When I was young, I ___ run for miles. (could/must) — could
You ___ have locked the door — I clearly remember doing it. (needn't/must) — needn't have
He ___ be lying; his story keeps changing. (must/can) — must
___ you help me with this? (Could/Must) — Could
We ___ obey traffic rules. (should/might) — should
You ___ smoke here — it's strictly prohibited. (mustn't/needn't) — mustn't
I ___ come if you don't want me to — it's optional. (needn't/mustn't) — needn't
She ___ have passed — she studied so hard. (should/must) — should