The nuances of these requests are not far from their English counterparts, actually.
Please write more clearly.
This is a direct request, but isn't rude by any means.
Would you write more clearly (for me)?
This, just like in English, is less direct and therefore showing a bit more respect to the listener.
Wouldn't you write more clearly (for me)?
In this form it feels more like an invitation than a question but is very similar to the previous.
In most cases I would use the first. It has a pretty broad usage. If I were feeling the need to be more polite (maybe I'm speaking to a teacher I don't know well) then I would use the second. If I were inviting a superior to do something for me or come somewhere I would use the third. Knowing which one to use often comes down to having lots of practice with social situations and a good grasp on context.