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Au Clair de la Lune
The website First Sounds is an interesting place to find the very beginnings of sound recording.
The project is ran by 'audio historians, recording engineers, sound archivists, scientists, other individuals, and organizations' and their main goal is to let people hear the sounds recorded at the beginning of technology.

Recently they have managed to restore the sounds found on four phonautograms. Phonautograms are sheets of paper that have been blackened by oil lamp smoke and would be made with an phonautograph.
Follow up:
One sheet contains a recording of the French song 'Au Clair de la Lune' that is seen as the oldest music performance ever recorded; on the 9th of April 1860. The quality is what you would expect from paper blackened by an oil lamp but this is still kind of cool to listen to. It is older then the recordings by Edison.
You can listen to the recording of Au Clair de la Lune on FirstSounds.Org here.
I have provided a YouTube clip of a modern version below for reference.