songs that may have been sung while a woman is laboring?

From this, I take it she is not talking about joke songs, parodies, songs composed about the topic of giving birth, or songs that happen to have been in one's head while giving birth. The language implies some kind of "traditional" or customary songs.

What I can't figure, Doulalady, is what linguistic or cultural background you are looking for. Is it English-language speaking culture that you need?

If so, I can't think of anything traditional to the Anglo culture (though I'd be happy to be corrected)...in fact, English culture seems to have little, in recent centuries, in the way of (traditional) "life cycle" songs tied to events and rituals and functions like that. (Even chanties had to be borrowed by the Anglos from the African-Americans :P )
However, look outside that culture, and examples abound. I know a lot about the songs from Punjab (~Northern India), in which case there seem to be / have been customary songs attached to each stage (of many) in weddings, as well as births, deaths, etc. None that I've heard of to be sung during the actual labor though (the songs are sung just after the birth). But during-labor songs—after a quick internet search—look like they might be there in Laotian culture, in Navajo, and more.

My answer in short is, yes, there are many such songs, though I have not heard of anything in English. Good luck on the search!