Question

Can ninetales learn any fairy move?

Answer 1

Yes, and that's...
Baby Doll Eyes
Thats it.

Edit: Now that Sun/Moon is released and the database seems up to date I'll add some more fairy type-moves that Alolan Ninetales can use:
- Dazzling Gleam (lvl up/evolve/TM/)(in SwSh; TR)
- Charm (egg move)(in SwSh; TM)
- Moonblast (egg move)

Edit 2: Updating to include SwSh:
- Draining Kiss (TR)
- Misty Terrain (TR)

Source: http://pokemondb.net/pokedex/ninetales

Comments

Note that there is a very high probability that Alolan Ninetales will learn several Fairy type moves in Pokemon Sun & Moon, owing to it's Ice/Fairy typing.

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I'll edit it when the movesets in Sun/Moon are fully revealed. Thanks!