> Presumably, in a feudal society commoners do not have surnames at all

There's confirmation of this in the short story [The Sworn Sword][1] (published as part of [A Knight of the Seven Kingdoms][2]).

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". 

<sub>*Weirdly, two brothers were both called Wat, no other distinguishing names. I genuinely don't understand how that works.*</sub>

> so it would be strange if their bastards did.

Commoner bastards often never even find out who their parents are, at all. 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 likely to also be a bastard, left to fend for himself.


  [1]: http://awoiaf.westeros.org/index.php/The_Sworn_Sword
  [2]: http://awoiaf.westeros.org/index.php/A_Knight_of_the_Seven_Kingdoms