[user104658]

I'm pretty sure the "bastard title" (Snow, Rivers, Sand etc) is specific to the first generation offspring of a house name (Stark, Bolton, Martell). All of the ones we've seen have been anyway... And logically, if it became an inherited surname, there would be loads of little Snows running around after a few hundred years... All you'd need is one bastard ancestor.

So I'd think of it more as a title / description than a surname... Specifically so that people know you as "whoever's child". From what we've seen it carries far more status than being a commoner, whilst deliberately being kept at arm's length from the noble line. They'd be much more willing to accept Jon having a real title if they knew he was of full noble blood, though technically, he would be Jon Targaryen not Jon Stark, as it's his mother who was a Stark.

My guess for the children of bastards is that they would just take the mother's family name?