Modal Verbs of Permission

Use the modal verbs of permission, 'can,' or 'could,' to indicate whether someone has permission to do something or not.

Use 'can' to say that someone IS allowed to do something and 'cannot' or, 'can't,' to say that they do NOT have permission (are not allowed) to do something.

Example:
Students can register for fall classes beginning next week.
Children under ten cannot use the pool without adult supervision.

We can also use 'could' to say that someone was allowed to do something in the past. We use 'could not' or, 'couldn't,' to say that they were NOT allowed to do it.

Example:
We could go to any shop in the mall we wanted to.
Both staff and students could use the ice rink.
We couldn't study in the library after 6 pm.

We use 'be allowed to' when talking about permission but not in the sense that you are asking for it or granting it. For instance, we would say "I was allowed backstage after my third attempt." Or, "you are not allowed to use your calculator on your math's exam."

Example:
"Mr. Smith, may I go to the restroom?"
"You may go as soon as you have finished your exam."
Retailers may not sell items below the manufacturer's price.

However, we use the modal verbs of permission 'can' when we are giving permission to others.

Example:
"You can borrow my book if you want, Diane."
"You can go leave work now, I've got it covered."
"Tom can go with you."

When refusing permission to someone, we use negative forms of the modals of permission; cannot = can't, or will not = won't, or shall not = shan't.

Example:
“Can I leave early?- “No, you can't!”
“Ill go out on Saturday.” - “You will not!”
You shan't get any money until the work is satisfactorily finished.

Modal Verbs of Permission = politeness, respect

In many languages, the third person is used even when talking to an individual as a sign of respect and formality. In the English, however, we don't have such a luxury and resort to voice inflection and modal verbs to show politeness or respect to someone unknown to us or someone older than ourselves.

'Can' is less formal and more direct than 'could.' When asking in a simple or direct way we use 'can.'

Can I ask a question?
Can we have something to drink? We are thirsty.

When being polite, or wanting to show respect, 'could' is a better choice than can as it is a more polite form than 'can.'

Could I ask you a question, please?
Could I try this coat on?

'May' and 'Might' can also be used when asking permission but are much more formal.

May I use your restroom, please?
Might I ask what your name is?

'Might' is rarely used in this way as it's considered old fashioned usage, however, educated and polite people still use modals to indicate politeness and show respect and I, for one, am in favor of this kind of usage.