The well-known jazz bandleader Tommy Dorsey recorded a piece in the 1940s called "My dog has fleas". I wonder if it comes from that - I've not been able to find a recording. The phrase seems to be widely offered as a mnemonic as if it were self-evident, and perhaps the song was once so widely known that it would need no explanation.

Nowadays it seems you have to learn the tune first, which rather misses the point. Or is this another UK/US thing - is the tune, if not the origin, of "my dog has fleas" still well-known? I have to confess that I'm not familiar with "Great green gobs of greasy grimy gopher guts" and "On top of spaghetti" either.