Silverwing
The queen's dragon, who carried Alysanne to the Wall and lived through the war that killed nearly all the others.
The beast itself
Silverwing was pale silver over most of her body with wing membranes of a light, warm pink that turned rose-gold when she flew with the sun behind her, and she was, by common consent, the prettiest dragon of the long peace without being the showiest. She grew slowly and steadily to a good size without ever becoming one of the monsters, and she had a smooth, economical way of flying ÔÇö long glides, few wingbeats ÔÇö that let her stay up for hours and cover the whole of a kingdom in a day.
Of all the dragons in this collection she was the easiest to be near. Silverwing was even-tempered to a degree the dragonkeepers found faintly suspicious, curious about people rather than contemptuous of them, and content to be approached, touched and fed by hands she did not know. She made a low chirring sound when she was pleased that several chroniclers recorded because they had never heard a dragon make it. She was also, and this is the part that stayed in people's mouths, capable of simple refusal ÔÇö there were places she would not go, and no amount of urging from the woman on her back would move her.
Her first rider: Alysanne Targaryen
Alysanne claimed her young and flew her for a lifetime, and the two of them are responsible for a strange fact: the Targaryens were never more popular than in the decades when a dragon might land in your village and the queen would climb off it and ask the women what they wanted. That was the work. Silverwing carried Alysanne to the Vale, to the North, to the west and the Reach and the Riverlands, on progress after progress, and everywhere she came down the local lords discovered that the crown had opinions about how their daughters were being married off.
The most famous of those journeys took her north to the Wall, where Alysanne was well received, gave the Night's Watch a new castle and got the New Gift out of a Stark for nothing. It is also where the story everyone repeats happened: Silverwing would not go beyond the Wall. Three times she was put at it and three times she turned aside, and Alysanne, who was not a superstitious woman, wrote about it afterwards in a way that suggests it bothered her considerably.
Claimed by: Ulf White
Long after her rider, Silverwing lived unridden on Dragonstone, still perfectly willing to be handled, which made her the obvious prize when the blacks put out their call for anyone with the blood who fancied their chances. Ulf White was a man-at-arms of loose habits and looser talk, fond of drink, fonder of boasting, and by no means the most promising figure who came forward. He got the gentlest great dragon in the world, which is roughly what he deserved and considerably more than he could handle.
The pairing was a mismatch in the plainest sense: a patient, tractable animal carrying a man with no discipline and no cause he could name. Ulf flew her because it was better than not flying her, drank between engagements, and made no secret of the fact that his loyalty was for hire. Onlookers found the sight of the Good Queen's dragon under that particular rider hard to stomach, and said so, and were ignored.
What the maesters dispute
The refusal at the Wall is the most argued-over anecdote in dragon-lore, and the arguments run in three directions. Some say the beast simply balked at the cold and the height and the wind coming off the ice, which is dull and may be true. Some say the Wall was raised with something in it that dragons can feel and men cannot, which is thrilling and unprovable. Some say Alysanne, who had already been told the North was not her business, invented the whole thing as a graceful way of turning back. The Citadel prefers the first explanation and cannot get anyone else to.
What became of it ÔÇö if an ending is written, it is here. Spoilers.
Silverwing survived the Dance, one of only a handful to do so, and lived wild around the Red Lake for years afterward. She is part of the reason the dragons' extinction took another generation rather than happening all at once.