Determine how you want your saddle displayed. You have two basic options for saddles.
You can affix a wall-mounted saddle rack to your wall and place the saddle on it, which will be the most stable way to display the saddle without damaging it. The other option is to stretch the saddle out against the wall and physically attach it to the studs by driving screws into the wall through the leather on the stirrups and flaps.
Use an electric stud finder to locate the studs in your walls. The studs are the support beams and strongest points in your walls.
Due to the saddle's weight, it will have to be attached to the studs regardless of whether it's on a rack or put straight on the wall.
Use screws to attach your saddle rack to the stud in the wall. Use your measuring tape and level to ensure the rack is positioned correctly and that the saddle will not tip to one direction or the other.
If you plan on attaching the saddle directly to the wall, mark places on multiple studs where the saddle can be attached. Hold up the saddle on the wall to line up the thinner areas of the saddle (flaps, stirrup fenders) with the studs.
Screw the saddle directly into the wall, ensuring all the screws go completely through the saddle and into the stud. You will need two to three screws per side to keep the saddle upright and secure.
Distribute the screws through both the flaps and the stirrup fenders.