Well, I think that Greensleeves is a corruption of a title from a Gaelic language. "Slieve" means mountain, and the word is used in quite a few song titles.

To the person who thought it referred to the sleeves of women working in the fields: Don't you ever garden? When you work in a field, you wind up covered with dirt, not green stuff.

I read in an authoritative place, but I forget which, that Greensleeves appeared in the time of Elizabeth I and Shakespeare. So forget all those myths about Henry 8. Eliz had at least one Irish harper at her court, and maybe he brought the tune to England.