Fleet was he when he did fly
down the road under the sky.
Friend was he to both dwarf and elf
used when speed and courage demanded more of self.
Arod?
Arod was a light horse of Rohan. It bore a Knight of Rohan in Éomer's Éored that engaged the Orcs outside Fangorn. His rider was lost during the skirmish, but Arod himself survived and remained with the rest of the éored on their way back. In the Wold, Éomer gave him to Legolas.
note - I never went back but suspect the answer is “Arod”. The horse that bore both Legolas and Gimli through the Paths of the Dead. (Scott Baltes)
Failed: Shadowfax, Gwaihir
Tolkien Gateway: Arod
Content contributed by Jordi Chalmeta & Scott Baltes