Could you write your name?
Would you write your name?
When you are asked to do this, are there any situation in which you hear weird if either is used (but the other sounds pretty natural .)
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Both are very commonly used, but "would you" is more grammatically correct in this case.
Technically, the first question is asking if you are able to write your name, while the second asks if you are willing to write your name. If we assume that the speaker has no reason to doubt the other person's ability to write their name, then "would" is the better word choice.
The most proper way to use these words is to use "could" if you're not sure if the person is able to do something. For example, "Could you take care of my cat next weekend?". In that case, I am not sure if you're going to be able to - maybe you are going to be out of town or busy with something else.
If you know the person is capable of doing what is asked, then it's better to use "would". For example, "If you're going to the kitchen, would you get me a glass of water?" In that case I know full well the person I am asking is capable of doing what I ask, but I'm inquiring about their willingness.
That said, however, both are very common and used almost interchangeably in common speech, to the frustration of pedantic people everywhere. You can easily get away with using the "wrong one" and most people wouldn't even notice.