The Stable Saddle is an item from The Legend of Zelda: Breath of the Wild. It is the default saddle that Link receives upon registering a horse at a stable for 20 Rupees. Unlike most saddles, it does not appear in the key items menu and can only be seen in the saddle selection menu when switching saddles at stables. Stable Saddles can also be seen stored on racks inside various stables though these are simply for ambiance that shows where Stable Saddles are stored by Stables.
The only horses that do not receive one upon registrationare the Giant Horse and Epona which receive special Bridles and Saddles unique to them. The White Horse receives one upon registration by default as it is the only special horse that allows its mane and equipment be changed. Any saddle equipped to the White Horse will be replaced with the Royal Saddle when speaking to Toffa during "The Royal White Stallion" after taming and registering the White Horse, however its gear can be changed provided maximum bond is reached. Interestingly the Stable Saddle, Royal Saddle, and Giant Horse's Saddle do not require Link to be fully bonded to them to be equipped though the Giant Horse's Saddle and Stable Saddles are equipped automatically upon registration and the Royal Saddle is equipped during a Side Quest. Epona has maximum bond automatically thus it doesn't apply to her special saddle.