Legolas killed the Winged Messenger's steed

‘The victor would emerge stronger than either, and free from doubt,’ said Gandalf. ‘But Isengard cannot fight Mordor, unless Saruman first obtains the Ring. That he will never do now. He does not yet know his peril. There is much that he does not know. He was so eager to lay his hands on his prey that he could not wait at home, and he came forth to meet and to spy on his messengers. But he came too late, for once, and the battle was over and beyond his help before he reached these parts. He did not remain here long. I look into his mind and I see his doubt. He has no woodcraft. He believes that the horsemen slew and burned all upon the field of battle; but he does not know whether the Orcs were bringing any prisoners or not. And he does not know of the quarrel between his servants and the Orcs of Mordor; nor does he know of the Winged Messenger.’

‘The Winged Messenger!’ cried Legolas. ‘I shot at him with the bow of Galadriel above Sarn Gebir, and I felled him from the sky. He filled us all with fear. What new terror is this?’

‘One that you cannot slay with arrows,’ said Gandalf. ‘You only slew his steed. It was a good deed; but the Rider was soon horsed again. For he was a Nazgûl, one of the Nine, who ride now upon winged steeds. Soon their terror will overshadow the last armies of our friends, cutting off the sun. But they have not yet been allowed to cross the River, and Saruman does not know of this new shape in which the Ringwraiths have been clad. His thought is ever on the Ring. Was it present in the battle? Was it found? What if Théoden, Lord of the Mark, should come by it and learn of its power? That is the danger that he sees, and he has fled back to Isengard to double and treble his assault on Rohan. And all the time there is another danger, close at hand, which he does not see, busy with his fiery thoughts. He has forgotten Treebeard.’

The Two Towers, Book Three, Chapter 5: The White Rider.