The horse ridden by Gandalf in The Lord of the Rings is Shadowfax. He is one of the great mearas, a remarkable race of horses from Rohan known for intelligence, speed, endurance, and a close bond with their riders.
Shadowfax originally belonged to Théoden, king of Rohan. Gandalf receives him after escaping from Orthanc and later riding to Edoras. The horse becomes essential to Gandalf’s journeys, carrying him enormous distances with extraordinary speed and responding to commands without needing a bridle or saddle in Tolkien’s account.
The name Shadowfax evokes speed and darkness, fitting a horse described as swift as the wind and pale or silver-gray in appearance. Gandalf rides Shadowfax during major events, including his journeys between Rohan, Minas Tirith, and the battle against Sauron.
The distractors belong to other characters or stories. Brego is Théodred’s horse in the films, Bill is Samwise Gamgee’s pony, and Snowmane is Théoden’s famed horse. Shadowfax is therefore the single canonical answer.