Originally Posted by
Rollo
-Stark Bran Stark (with Rickon as his heir). No Stark heir thereafter.
-Arryn Robert Arryn (with Harold Hardyng as his heir). It is said if Robin were to die, Harold would take the Arryn name. Otherwise the Arryn name ends with Robert.
-Baratheon Shireen won't pass on the name. Tommen Baratheon needs a Baratheon heir before he dies (assuming he needs to die to fulfill Maggy the Frog's prophecy). Otherwise the Baratheon name dies.
-Lannister Martyn Lannister is heir, followed by distant cousins. It is unknown if Damon is still alive, but his son (Damian) and grandson (Lucion) are, and they would follow Martyn. Jaime could jump ahead if decreed by the King.
-Martell Arriane Martell is heir to Doran, followed by Trystane. Neither have an heir.
-Targaryn Aegon Targaryen and Daenarys Targaryen. Technically Aegon is the rightful heir to the family name being the Mad King's grandson. Even if he truly isn't a Targaryen, he's using the name and has backing (similar to Tommen passing on the Baratheon name).
-Tyrell Willas Tyrell is the first heir of Mace Tyrell. Followed by his younger brother Garlan Tyrell. Loras is a kingsguard and gave up his inherited titles (he is also near death). Third in line is Luthor Tyrell (descended from Mace Tyrell's uncle).
Houses Tyrell, Martell, and Lannister appear to be in good shape. Houses Stark, Baratheon, and Arryn appear to be endangered. The Targaryen line was assumed to be dead, but is coming back in full vengeance. It's a perfect flip from Robert's Rebellion.