There are 4 types of living mounts:
When you use a Saddle on a tamed or bought mount it will become a statue in your backpack, which can be used to mount your horse. When you dismount a saddled mount will return to your backpack as a statue. Saddles do not currently work on Giant Beetles.
Summon Ethereal Mount
Tamers who have a high enough taming skill can tame a Dread which is able to be used as a mount and does not require a saddle.