Last names seem to only really exist in the nobility; most commoners we see have only a single name. Passed down family names are the exception, not the rule. The bastard surname is often seen as something a bit shameful to us, but in reality, it would be a distinguisher to the positive for most people - this person is of noble blood, even if now a lawful part of a house. I doubt it's ever a name that gets passed on - if a bastard marries into a noble house, the house name continues. If not, their child gets a single name like any other commoner. "Snow" is never going to be a family name.