Modal verbs (must, should, could, might, would, can) express attitudes — necessity, advice, possibility, ability. Modal perfect (modal + have + past participle) talks about the past.
The basics
must = strong necessity. should = advice. could = possibility / past ability. might = weak possibility. would = imaginary / polite request. can = ability / permission.
Modal perfect — talking about the past
must have + V3 = certain it happened. might have + V3 = possibly happened. could have + V3 = possibly happened (or was able). should have + V3 = it didn't happen but it would have been better. shouldn't have + V3 = it happened but it was wrong.
- She might have taken her medicine. (we're not sure)
- She shouldn't have taken her medicine. (she took it — but she was wrong to)
- We could have spent the weekend there if only we had brought food.
- The ancient boat might have been carrying jewellery up to ten metres long.
Modal incorrect uses (exam trap)
'We only use (must) when we need to give an instruction very strongly.' This is INCORRECT — must has many uses (necessity, deduction, etc.), not only strong instructions.
Don't have to vs mustn't
Hasn't / doesn't have to = no obligation. Mustn't = forbidden. (Don't confuse them — this is a classic exam mistake.)