Formal:
* Kindly: "Kindly provide me with your contact information."
* Would you be so kind as to...: "Would you be so kind as to open the door for me?"
* May I request...: "May I request that you send me the document?"
* If you would be so gracious...: "If you would be so gracious as to lend me a pen."
* I would appreciate it if you would...: "I would appreciate it if you would pass me the salt."
Informal:
* Could you...: "Could you pass me the remote?"
* Would you mind...: "Would you mind closing the window?"
* Do you mind...: "Do you mind if I sit here?"
* Can you...: "Can you help me with this?"
* Thanks: "Thanks for helping me with this."
* Appreciate it: "Appreciate it if you could do that."
More specific:
* If you don't mind: "If you don't mind, could you check my work?"
* If possible: "Could you send me an update, if possible?"
* If you'd be so kind: "If you'd be so kind, could you give me a hand?"
* Would it be alright...: "Would it be alright if I borrowed your phone?"
Emphasizing politeness:
* I would be grateful if...: "I would be grateful if you could answer this question."
* It would be a great help if...: "It would be a great help if you could proofread this."
The best option ultimately depends on your specific situation and the tone you want to convey.