From the Encyclopedia of Arda:
A mighty horse of Rohan, the chief of the Mearas, tamed by Gandalf and reluctantly granted as a gift to him by King Théoden of the Rohirrim. Shadowfax is thought to have passed West over the Sea with his master
Notes
1 Tolkien never states explicitly that Shadowfax sailed into the West. In his Letters, though, he gives such a wealth of clues and hints, we can state with near-certainty that the great horse sailed on the White Ship with his master Gandalf. (See The Letters of J.R.R. Tolkien No 268, dated 1965).
2 Fax is an extinct word for 'hair' or 'coat', now only surviving in a few place- and family-names, like 'Fairfax'. In fact, Tolkien has anglicised the name 'Shadowfax' somewhat from its original form; in the Old English language representing the tongue of the Rohirrim who bred him, he would have been called Sceadufax or Scadufax.
