Given the scenario Diane posted:

ethically the rights should belong to the singer/songwriter; but unless the author can prove that, they are SOL, legally.

My opinon, ymmv


Another, (to my mind more interesting) scenario:

A field collector records a song from a source that the collector can find no record of being known previously. All evidence points towards the source having learned it as a young child from his great-grandaddy who died 70 plus years ago.

Where would "intellectual property" rights lie?