Presumably, in a feudal society commoners do not have surnames at all
There's confirmation that regular peasants have no surnames in the short story The Sworn Sword (published as part of A Knight of the Seven Kingdoms).
I'm being careful to not include any spoilers since many people won't have read it yet...
There's a scene where two knights are interacting with a group of peasants. Several of the peasants have the same name, and it gets very confusing, so the knights decide to assign them surnames based on the crops grown in the villages they come from (for example, I think there was a Wat Barleycorn).
The peasants are very excited at being given what they call "Lord's names". Clearly, surnames are normally for nobles only.
Weirdly, two brothers were both called Wat, no other distinguishing names. I genuinely don't understand how that works.
so it would be strange if their bastards did.
Commoner bastards often never even find out who their parents are, at all, and they certainly don't have surnames, even if they're knighted, unless they earn one by being granted lands and a household title. This is also confirmed in the same story. Small character-development spoiler:
Dunk discovers a high-born friend has anti-bastard prejudices, and chastises him, explaining that he himself never knew who his parents were, and therefore was most likely a bastard, left to fend for himself.
As a commoner who presumes himself to be a bastard, he doesn't take his region's bastard surname, and instead, on becoming a knight, invents himself a title based on attributes ("Ser Duncan the Tall"). Other knights similarly lacking surnames use attributes like where they're from, e.g. Ser Arlan of Pennytree.
If there's a dispute about surnames, it seems like the benefit of the doubt is given in as afar as people are granted the bastard surname if it's likely they had one noble parent, even if they can't prove it.
For example, there's a character in The Mystery Knight whose mother was a prostitute and who never knew who his father was. He gives himself the surname of the famous, long-dead knight from an extinct family who he believes to be his father (who is too dead to argue), but he's announced using the generic bastard surname of his region. Arguably, this is a courtesy since there isn't any known proof at this time that his father wasn't a commoner, but he takes it as an insult.