Questions with might, may, and can

 Might I... is used to ask permission or to make a suggestion in a very formal and polite way.

• Might I see your credit card, please?
• Might I recommend the grilled swordfish?
• Might I remind you that this is a library? (an extremely formal and critical way of asking someone to be quiet)

Might I ask... is used to ask permission to ask a question, so it is very polite indeed:

• Might I ask how you heard about our company?

There are two other verbs that can be used for the same function as might, but without conveying such extreme formality: may and can. May I is similar to might I and is used in many of the same contexts, but it is less formal. It is also much more common than might I.

• May I see your driver’s license?
• May I help you?
• May I offer you something to drink?

Can I is the least formal of the three, as shown in these examples:

• Might I call you by your first name?  (very formal)
• May I call you Elizabeth?                   (formal)
• Can I call you Liz?                             (infomal)