Sometimes it is difficult to choose between two modal verbs that best fits in the situation. Here are a few special cases of modal verbs that can help you choose the correct modal verb for the situation.

SHALL v/s WILL

As the contracted form of both these modal verbs is 'll, the difference between these two modal verbs is sometimes hard to see.

Some people say that no one uses "shall" anymore, but that isn't true at all!

The basic pattern is that British English uses "shall" far more often, and in American English, it shows politeness.

If you want to know more, here is the traditional usage pattern:

First option: basic statement of fact

I/We + shall

For example:

I shall arrive before you.

You/He/She/It/They + will

For example:

She will arrive before you.

Second option: promise, command, or strong statement
I/We + will

For example:

We will do everything we can to find your dog.

You/He/She/It/They + shall

For example:

They shall not rest before finding him.

"Shall" is also more likely to be used in legal documents.

For example:

The tenant shall pay rent by the 5th of the month.

SHOULD v/s WOULD

For the most part, these two verbs are used in very different ways.

"Should" is used to talk about obligation and give suggestions.

For example:

They should take more vacation time.

(Giving a suggestion)

You should do your homework every night.

(Explaining an obligation)

"Would" is used to discuss habitual action in the past.

For example:

As kids, we would play outside all summer.

However, "should/would" do have a little overlap. It roughly corresponds to the past tense of "shall/will," as discussed above.

For most subject pronouns, "should" shows a stronger idea of command, promise, or assertion. For I and we, the opposite is true - "would" is stronger.

For example:

I should think they are telling the truth.
is less certain than
I would think they are telling the truth.

If they did their homework, they would do better.
is not as strong as
If they did their homework, they should do better.

As with "shall/will," "should" is used more often than "would" by the British, and can more often be found in legal documents or formal language.

CAN v/s COULD

In general, "could" is the past tense of "can," especially when making statements about possibility.

For example:

You can find freshly-baked bread at the market. (now, in the present)

You could find freshly-baked bread at the market. (at some time in the past)

"Could" is also used to show a future possibility that is not totally certain.

For example:

If the bakery moves in, we could buy freshly-baked bread on our street!

In requests for permission, or granting of permission, "could" is more formal than "can."

For example:

Can I borrow a pencil?
is less polite than
Could I borrow a pencil?

(request for permission)

You can take the subway
is less formal than
You could take the subway.

(granting permission)

"Could" is also a polite way to ask someone to do something.

For example:

Could I have more water?
is more polite than
Can I have more water?

Summary

SHALL v/s WILL: The basic pattern is that British English uses shall far more often, and in American English, it shows politeness. Shall is also more likely to be used in legal documents.
SHOULD v/s WOULD: For the most part, these two verbs are used in very different ways. Where they do overlap, they correspond to the past tense of shall/will
CAN v/s COULD: In general, could is the past tense of can, especially when making statements about possibility. Could is also used to show a future possibility that is not totally certain. In requests for permission, or granting of permission, could is more formal than can. Lastly, could is a polite way to ask someone to do something.