In Game of Thrones/A Song of Ice and Fire, bastards have the last names such as Snow, Sand, Flowers, Hill, Pyke, Rivers, Stone, Storm, Waters. In the event that they are never legitimized what happens when these bastard children become adults? Do they and their offspring forevermore hold these last names or as they grow up do they invent a name for themselves? Would that even be allowed?

Edit Pt2: What happens to unacknowledged bastards? Do they get to make up names too?