When you can't rent a horse, they aren't in the yard, it means that it is Horse Station World Change.
Wild Horses spawn is every 3h from ss. Then 0 to 3 monsters spawn each time, this is random.
bestiary gives
10 charm points when you killed 5 horses.
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You can tammed it by
Sugar Oat or
Music Box to get mount and achievement "Lucky Horseshoe".
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Btw you should make bestiary for all others horses, it's
5points for each, you must kill 250 but it's easy and don't waste your stamina.