It is quite common to have an introductory verse in which the singer says s/he spied or overheard a woman lamenting. That would indicate that a man could sing the song telling of a woman's plight.

there are a couple of academic studies of the songs of women who dressed up as sailors and soldiers (these songs sometimes were based on true stories), but that's a different category isn't it? And I'm sure there must be anthropological studies of men and women's roles and repetoires in singing in different societies.