It kind of depends on what you mean by song.

The sheet music in front of me, for example, is something very specific. The words are James Clarence Mangan's poetic translation, with the title, "My Dark Rosaleen." And in this case, the music was composed, for voice and piano, by the Anglo-Irish Alicia Adelaide Needham, who included this song in her multi-song cycle, "A Bunch of Shamrocks." Needham was grounded in classical music and had studied composition formally.

This is the very song recorded on 78 RPM by John McCormack. The original song was written for low voice; McCormack, for obvious reasons, had the key transposed considerably higher for his tenor voice, and the song was arranged for a studio orchestra rather than the piano for which the accompaniment was originally written.

"My Dark Rosaleen," the Mangan/Needham setting, is now in the public domain, copyrighted as it was in 1897 by Boosey & Co (not yet Boosey & Hawkes at that point in history), in London.