Snow is the surname for bastards north of the Neck (I: 12)
Bastards are said to grow up more swiftly than other children (I: 45)
It is not unexpected for noblemen to have bastard children (I: 54)
It is not typical for a noble to bring his bastards home and raise them with his own children. It's more usually expected that he will see to the child's well-being to some degree (I: 55)
The baseborn have few rights under the law and custom, when it comes to claims (I: 267)
Stone is the bastard name in the Vale (I: 309)
Flowers is the bastard name in the Reach (I: 309)
Each of the Seven Kingdoms have bastard surnames decreed by custom, although only noble bastards receive them (I: 309. SSM: 1)
Rivers is the bastard name in the riverlands (I: 541)
Pyke is the bastard name on the Iron Islands (I: 654. III: 364, 550)
Storm is the bastard name in the stormlands (II: 119)
A bastard may inherit if the father has no other trueborn children nor any other likely kin to follow him (II: 185)
Hill is the bastard name for the westerlands (III: 10)
Blackfyre was a name carried by a bastard of Aegon IV and his sons, but does not seem to have been a bastard name commonly used for Targaryen for Targaryen bastards, as his half-brother, whose mother was a Blackwood, used Rivers (III: 407, 521. TSS: 121-122)
Sand is the bastard name of Dorne (III: 431)
A bastard can inherit if he is legitimized by a royal decree (III: 521, 819. SSM: 1)
Aegon IV legitimized all his bastards, both the Great Bastards gotten on noble mothers and the baseborn, on his deathbed, and the pain, grief, war, and murder that wrought lasted five generations because of the Blackfyre pretenders (III: 521. TSS: 132)
It is rude to pry into the origins of a man's natural children (III: 766)
Men say that bastards are born from lust and lies, and so their nature is wanton and treacherous (III: 830)
Waters is the bastard name of Dragonstone and the King's Landing region (III: 929. IV: 120. SSM: 1)
Many noble bastards take the arms of their fathers with the colors reversed (TSS: 109. IV: 569)
Bastards whose parents are both of the nobility are not considered baseborn (TSS: 132)
The trueborn children of a bastard might change their surnames to show their legitimate nature. For example, a legitimate son of a Waters might change their surname to Longwaters (IV: 120-121. SSM: 1)
If two bastards from different regions married (such as a Snow and a Rivers), their offspring would probably take the name of their father (SSM: 1)