• Can I ask a favour?
• Can you do me a favour?
• Could you do me a favour?
• Could I ask you to do me a favour?
• I was wondering if I could ask you a favour. (indirect question in the past continuous tense)

http://www.bbc.co.uk/worldservice/learningenglish/radio/specials/142_requests_offers/page9.shtml

• Can I + infinitive...?
• Could I + infinitive...?
• Would I be able to + infinitive...?
• Is it OK/alright if I + present simple...?
• Would it be OK/alright if I + present simple(less formal)/past subjunctive...?

• I hope you don't mind me mentioning this but...
• I wondered if we could have a quick chat about...
• Would you mind (not) + gerund...?
• It would be good if you could...

• I would like to book an appointment...please.
• I need/want to book an appointment...please.
• Is it possible to book an appointment...please?
• May/Can I book an appointment...please?

• Would you mind + gerund...?
• Are you OK to + infinitive...?
• Is there any chance you could + infinitive...?
• I would appreciate it if you could...
• It would be great if you could...
• If you could..., that would be great.