When did voicemail first appear? You could argue it was in 1877.
On November 29, 1877, Thomas Edison demonstrated his invention, a hand-cranked phonograph that recorded sound on grooved metal cylinders. Edison shouted verses of "Mary Had a Little Lamb" into the machine, which played back his voice.
Soon after, Edison invented voicemail, in the sense that he sent a phonograph to a friend in England and they communicated by sending recordings back and forth over the Atlantic.
... David Ferris, with thanks to Mel Mandell