2.marginals
Four marginal modal verbs exist in English: need, dare, ought to and used to. These modal verbs do not have all the features of central modals. Used to takes the to-infinitive and is only present in the past tense (the contracted negative form usedn’t is possible). Ought to has the to-infinitive as well (ought not to can be contracted to oughtn’t). Dare and need may function either as modals (with bare infinitive, without inflections) or as full verbs (with to-infinitive and inflected -s, -ing or past form).

3.modal idioms
Modal idioms combine auxiliary verb and infinitive or adverb. They have no nonfinite form and are the first verb in verb phrase. The most frequent idioms are had better, have got to, be to and would rather.

4.semi-modals
Semi-modals are made by the means of primary verbs have and be. They have nonfinite form and are preceded by auxiliary verbs. Semi-modal phrases like have to, be supposed to, be likely to, be going to, be due to, be bound to, be about to, be able to, be obliged to, be willing to are somewhat common in the language.

There is also a special group of fixed phrases called catenatives. Catenatives are not modals, but they have a similar meaning to that of modal auxiliary verbs considering the marking of modality. Happen to, appear to and seem to are all catenatives.

Meaning
Modal verbs have several meanings. They may express permission, possibility, ability, obligation, necessity, volition or prediction. One modal verb has more meanings and expresses different functions.

“I can hear what she’s saying to somebody.”(ability)
“Can I have a piece of paper, please?”(permission)
“I can pick you up from over here.”(possibility)

Each modal auxiliary can also have deontic (intrinsic) or epistemic (extrinsic) meaning. The first type refers to events that are controlled by humans (or other agents), whereas the second refers to events, states or judgments that are, or are not, likely to happen. Some modal auxiliaries overlap both the deontic and epistemic modality meanings.

“You cannot mark without a scheme. You must make a scheme.”(intrinsic)
“In other cases his decisions will seem more radical.”(extrinsic)