Basically, what I want to know is how historically accurate A Song of Ice and Fire is in this regard. Several characters in the series have said something along the lines of "bastards are born of lust and betrayal, and so are themselves prone to lust and betrayal as well. You can't trust them". Did people actually think like that?
I'm currently starting a paper about some bishopric bastards - though I'm not yet at the stage of sharing much information.
One book I came across though through recommendation of my professor is:
By the way: William the Conqueror was a Bastard, nicknamed by some "William the Bastard" - but it was apparently not too good for your health to say that to his face. Another prominent bastard in England is Robert of Gloucester, the oldest natural son of Henry I (so a grandson of William). But at his lifetime it was apparently more difficult to be an illegitimate child with high ambitions. Robert still became one of the most important political players in the time of the "Anarchy", being essential for the Angevin side of that civil war between his half-sister Mathilda/Maud (whom he supported) against their cousin Stephen.
It was really very much depending on who your father and mother was, and what they thought of Bastards - some Bastards could really climb high - say as military men or in the church. I don't have any more concrete examples from the top of my head than the aforementioned.
Much later - in the 18th century there was the case of Charles Eugene, Duke of Württemberg who took care of his many bastards. The numbers would probably even impress Robert Baratheon - so there are those cases just like in ASOIAF that many people within a country can claim royal blood through bastards in this part of Germany. So once more this is a case where George Martin was inspired by actual history.
Edit: Just as I post this I remember another bastard - or rather someone who was long assumed to be a bastard, which by now is not as certain anymore: Arnulf of Kärnten, who became king in Western Francia. It has long been assumed that he was illegitimate and therefore not recognized as a real Carolingian. As it turns out that might be just propaganda by his opponents, and he was actually born by a mother whose marriage was not recognized by the father's royal father at this moment. It's in German, but the text that brought this up is: