It occurs to me that 'singer' might mean one who singes - ie burns lightly - rather than one who just sings! Having said that, I'm not sure why one would want to do such a thing to a horse. Perhaps bristly parts of a horse's coat had to be so treated, a bit as a barber might do to someone today?
It could, on the other hand, simply be a mis-spelling for 'ringer'. Mares used to be ringed, which involved having special rings fitted - don't ask me how! - to prevent them being mated. It's hard to imagine this as an actual occupation, however.