The stables at Freehold (NJ) Racetrack had a couple of horrible fires over the years, too...lost a lot of prize horses.

Put the horse in its stable.

But in speaking of an all-inclusive complex I've always used and heard sstables, as in the Freehold Stables.

However, if you were speaking about a particular building in the complex, you might say "How many stables are in this stable?" Each berth is a stable, but each separate building can be a stable, too. Hmmm.