Permission, offers, etc...

To ask PERMISSION to do something we use can or could:
• Can/Could I take another biscuit?
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. For example, you might use couldn't where you expect that the answer is likely to be 'no', or where permission has been refused before:
• Can't/Couldn't we stay just a little bit longer? Please?
To give and refuse permission we use can and can't:
• Okay. You can stay in the spare room.
• No, you can't have another chocolate.
Notice that we prefer can/can't rather than could/couldn't to give or refuse permission:
• I'm sorry, no, you can't borrow the car tonight, (rather than ...no, you couldn't...)
In rather formal English, may (not) can also be used to ask, give or refuse permission, and might can be used to ask permission (e.g. 'Might I ask...?').
We use can (for the present or the future) and could (for the past) to report permission.
• Jim says that we can borrow his house as long as we leave it clean and tidy.
• He said we can/could use the car, too.
To report that in the past someone had general permission to do something, that is, to do it at any time, we can use either could or was/were allowed to. However, to report permission for one particular past action, we use was/were allowed to, but not could:
• Last century, women were not allowed to vote, (or ...couldn't vote.)
• Although he didn't have a ticket, Ken was allowed to come in. (not ...could come in.)
In negative sentences, we can use either couldn't or wasn't/weren't allowed to to report that permission was not given in general or particular situations:
• We couldn't / weren't allowed to open the presents until Christmas.
If we use the present perfect, past perfect or an infinitive, we use be allowed to, not can/could:
• They have been allowed to keep the Roman coins they found in their garden.
• She is unlikely to be allowed to travel on that airline again.
When we OFFER to do something, or offer by making a suggestion, we can use can or could:
• Can/Could I help you with your bags? • You can/could borrow my car if you want.
In offers that are questions we can also use shall or should:
• Shall/Should I phone for a taxi for you?
If we use could or should we sound less certain that the offer will be accepted.
We also use shall/should in questions that request confirmation or advice:
• Shall/Should I put these books over here? • Who shall/should I pass the message to?
We can use Would (you) like when we make an offer, but not 'Will...':
• Would you like me to get you some water? (not Will you like me...?)
In requests, too, we can say (I) would like..., but not '(I) will...':
• I would (or 'd) like an orange juice, («of I'll like...)
We can use should (with I or we) instead of would in requests like this, but this is formal.
Compare these ways of offering food and drink:
• What will you have to eat/drink? (not What would you have to eat/drink?)
• What would you like to eat/drink? (not What will you like to eat/drink?)