NOTE: Unicorns cannot wear barding.
Wild unicorns can wear saddles and gold and electrum horseshoes only, but they must be tamed or charmed first, and then branded to be able to change their equipment.
Wild unicorns can carry a rider and a passenger.
Vanity unicorns (the pink and purple ones) can wear saddles, but not horseshoes. You do not have to tame them, but they must be branded to be able to change their equipment.