The verbs will, would, shall, should, can, could, may, might, and must cannot be the main (full) verbs alone. They are used as auxiliary verbs only and always need a main verb to follow.

Will

Used to express desire, preference, choice, or consent:

I will take this duty.
Will you stop talking like that?
Used to express the future:

It will rain tomorrow.
The news will spread soon.
Used to express capacity or capability:

This bucket will hold two gallons of water.
This airplane will take 200 passengers.
Used to express determination, insistence, or persistence:

I will do it as you say.
Would (past form of will)

Often used in auxiliary functions with rather to express preference:

I would rather go shopping today.
We’d rather say something than stay quiet.
Used to express a wish or desire:

I would like to have one more pencil.
Used to express contingency or possibility:

If I were you, I would be so happy.
Used to express routine or habitual things:

Normally, we would work until 6 p.m.
Shall

Mainly used in American English to ask questions politely (it has more usages in British English). For the future tense, will is more frequently used in American English than shall.

Shall we dance?
Shall I go now?
Let’s drink, shall we?
Often used in formal settings to deliver obligation or requirement:

You shall abide by the law.
There shall be no trespassing on this property.
Students shall not enter this room.
Should (past form of shall)

Often used in auxiliary functions to express an opinion, suggestion, preference, or idea:

You should rest at home today.
I should take a bus this time.
He should be more thoughtful in the decision-making process.
Used to express that you wish something had happened but it didn’t or couldn’t (should + have + past participle):

You should have seen it. It was really beautiful.
I should have completed it earlier to meet the deadline.
We should have visited the place on the way.
Used to ask for someone’s opinion:

What should we do now?
Should we continue our meeting?
Should we go this way?
Where should we go this summer?
Used to say something expected or correct:

There should be an old city hall building here.
Everybody should arrive by 6 p.m.
We should be there this evening.