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Helping verbs, or auxiliary verbs, such as: will, shall, may, might, can, could, must, ought to, need are used in conjunctions with main verbs to express shades of time and mood.
There are other helping verbs, called modals, which are can ,could, may, might, must, ought to, should, shall, will and would.

-they do not change form for different subjects.
-no -ing form after a modal.
-you can't find -to after a modal, except ought to.

The shades of meaning among modals are multifarious and complex. There are several aspects of modality action.

OBLIGATION AND PROBABILITY

We can use should and ought to with a little difference in meaning when we are discussing obligation and probability.

Obligation:

· We can use these modals for giving advice or making a recommandation.
You are not very well. Perhaps you should / ought to see a doctor.
Nevertheless, we use should, not ought to, when we give advice with I.

· We can use either should or ought to for talking about a responsability or duty.
I should / ought to visit my mom more often.

Probability:

· When we want to say that something is probably true now or will probably be true in the future, we use either should or ought to.
I enjoyed the first episode of the Lords oh the Rings, so the next one should / ought to be good.

· Had better to, especially in spoken English, is used to say what the best thing to do is in a given situation .
It is cold. You'd better wear a coat. (This is stronger than should or ought to. We don't use it to talk about the past or to make general comments)

· Be supposed to is commonly used in spoken English to express a less strong obligation.

WILLINGNESS, LIKELIHOOD, CERTAINTY

Willingness:

· We use will when we talk about willingness to do something (in offers, invitations, requests or orders). Will not expresses reluctance or refusal.
· With would, we talk about general or repeated willingness in the past. Moreover, we use would for a past prediction or a prediction about a possible situation.
M. had walked a long way. He would sleep well that night.

Would like is less direct than want. It is a more polite way of asking for something that you want, or inviting someone to do something.
Would you like to visit the museum?

Certainty:

· We use will (won't) to indicate that we think a present or future situation is certain.
Ex: I won't go to Bayrou's meeting.

· To indicate that we think a past situation is certain, we use: will + have + past participle.
They will have reached home by now.

· To indicate that we think an unreal past situation is certain, we use: would have + past participle.
I would have been happy to see him, but I did not have time.

HABITS

· We can use will (for the present) and would ( for the past) to talk about characteristic behaviour or habits, or about things that are or were always true.
During the war, people would eat all kind of things that we don't eat now.

· Either would or used to + infinitive (which is common in informal English) are used to talk about things that happened repeatedly in the past, but don't happen now. However, we use used to but not could when we talk about past states that have changed.
The factory used to be in the city centre.

· We normally make questions and negatives with used to in spoken English with did.
I didn't use to like visiting the dentist when I was young.

· There is an other question form in more formal spoken and written English, with used to.
Used you to go to university with the Evans brothers? (quite rare now)

POSSIBILITY I

· In affirmative sentences, we use may (might) to say there is a possibility of something happening or being true. Might suggests there is less possibility.
This may / might be his last major speech before the election.
Can: when we talk about a more general possibility of something happening
Could: we are giving an opinion.

· In questions: To ask questions about the possibility of something happening, we use couldn't or the phrase be likely to.
Are you likely to be visiting Greece again this summer?

· In negative sentences: can't (couldn't): it is not possible that something is true.
May not (might not): it is possible that something is not true.

POSSIILITY II

In the past:

· May / might / could + have + past participle: We use this to say it is possible that something happened in the past.
Tom is late. He may / might / could have forgotten about the meeting.

· Might / could + have + past participle: something was possible in the past, but we know that it did not happen in fact.
The plan might / could easily have gone wrong, but in fact it was a great success.

· Might + infinitive: To talk about what was typically the case in the past.
During the war, the police might arrest you for criticising the King.

In the future:

· May / might + have + past participle: To say that by some time in the future, it is possible that something will have happened.
By next week, I may / might have completed my civilisation dossier.

· May / might + be + -ing: To say it is possible that is happening now or to talk about a possible future arrangement.
Malcolm is not in his office. He may / might be working at home today.

ABILITY

· With can't (for the present) and couldn't (for the past), we express the fact that someone or something has or doesn't have the ability to do something.
We can't afford to pay the bill.

· In general, we use be able to when we talk about a specific achievement rather than a general ability (particularly if requiries some effort)
He has now recovered from his injury and is able to drive again.

However, even when we are talking about specific occasions, we commonly use can or could with verbs of the senses (feel ,hear, see... ) and with verbs of "thinking" (believe, decide, remember...)
I can't decide where to go for my holidays.

· We don't use be able to when we talk about something that is happening as we are speaking, and before be + past participle.

NECESSITY

· In the present, we use must and have to to say that something is necessary.
You must be careful.

· There is a difference between must and have to.
We use must when the speaker decides it is necessary.
You must finish this today. (I'm telling you)

We use have to when the necessity is outside the speaker.
I have to finish this today. (The boss says so)

The negative form:

· We use mustn't to stop or forbid someone from doing something.
You mustn't walk on the grass.
We use needn't when it is not necessary to do something.
You needn't speak so loudly.

· We can use either needn't or don't have to when we say that it is unnecessary to do something. The meaning is the same.

PERMISSION, OFFERS...

Permission:

· To ask permission to do something, we use can or could. We use could to be particularly polite. If we want to put extra pressure on someone to give a positive answer, we can use can't or couldn't. We prefer can / can't rather than could / couldn't to give or refuse permission.
You can't borrow the car tonight.

In rather formal English, may (not) can also be used to ask, give, refuse permission, and might can be used to ask permission.
Might I ask..?

· To report permission in the past: To report that someone had general permission to do something (to do it at any time), we can use either could or was/were allowed to.
In the last century, women were not allowed to vote ( or couldn't vote)

To report permission for one particular past action, we use was / were allowed to, but not could.
Although he didn't have a ticket, Ken was allowed to come in ( not could come in)

Offer:

· When we offer to do something, we can use can or could.
You can (could) borrow my car if you want.

With could and should, we are less certain that the offer will be accepted.

We also used shall / should in questions that request confirmation or advice
Shall / should I put these books over here?

We can use would (you) like when we make an offer, but not will.
Would you like me to get you some water?
In requests, we can say (I) would like, but not I will.
I would like an orange juice (and not I will like...)
Should instead of would is possible in request like this, but it is rather formal.