～てくれませんかVS てもらいませんか

[Taiko99]

Which is more commonly used for asking for a request politely?

[Javizy]

A request can't end in もらいませんか. It would translate something like 'am I not going to get from you?'. In a もらう sentence you are the one doing the action of the verb, but since it involves another person, you have to ask if it is possible for you to do that action, i.e. the receiving of the object, and so use the potential form もらえませんか 'can't I get from you?'. It's important to note the direction, and the inferred subject/indirect object of the receiving verb.

彼女は私にくれる : She gives to me.
私は彼女にもらう : I get from her.

You probably already know that, but it's important to pay close attention to it in order to understand exactly what you're asking when you make a request. As far as I'm aware, there is no right or wrong answer to your question, it's just a case of emphasis. Much like with English, the situation will usually determine the most appropriate way of asking (how direct you want to be? do you want them to be able to decline easily?).

(私はあなたに)犬を洗ってもらえませんか : can't I receive from you the washing of my dog? / can't I get you to wash my dog for me?
(あなたは私に)犬を洗ってくれませんか : won't you do for me the washing of my dog? / won't you wash my dog for me?
(私はあなたに)犬を洗ってもらいたいんですが(洗ってもらえませんか) : I'd like you to wash my dog for me (so won't you wash it for me?). [another common way of asking someone to do something that they can more easily decline]