The most famous bastard sons of House Targaryen were the Great Bastards of Aegon IV in his deathbed, the King legitimized them all, causing five major rebellions and more than sixty years of troubles all the males had a personal Coat of Arms derived in part of the standard Targaryen three-headed dragon. I'm focusing my answer on bastard sons Aegon's answer has covered pretty much all the legitimate Targaryens, there is no point to duplicate information
