A horse statue with a saddle
A saddle was a type of harness typically applied to a horse or similar animal, designed to provide comfort for the rider.
Horse statues seen in the reliquary at the Vulcan monastery at P'Jem had saddles. (ENT: "The Andorian Incident")
Christopher Pike kept a saddle on display in his ready room on the USS Enterprise. After being transported back in time from 2381 to 2259 via the Krulmuth-B portal, Brad Boimler could not resist climbing astride it. When caught by Nyota Uhura, he claimed that he was appraising the "historical saddle". (SNW: "Those Old Scientists")
According to Captain Jean-Luc Picard, a saddle was a personal thing to a rider, and most serious riders had one of their own. He kept his own saddle on board the USS Enterprise and went to get it after seizing on Commander Calvin Hutchinson's mention of horseback riding as a means of excusing himself from the reception. He became trapped on the Enterprise during a Baryon sweep, and later used it as a means of throwing Devor off-guard when he attempted to attack him with a laser welder. Someone later put the saddle in a maintenance locker, where it was found by Lieutenant Worf. When he expressed surprise at Picard's possession of it, Commander Riker and Doctor Crusher playfully chastised him, as "anyone who is an experienced rider naturally has his own saddle." Picard expressed regret that he didn't have the opportunity to actually use it on a horse. (TNG: "Starship Mine")