Wiki - Shrinking & Stabling Pets
Once a player has a pet tamed there are a couple options as to how to store it. Regardless of the pet it will get hungry and restless over time, so if it is left on it's own it will eventually go wild.
The best way to store your pet is to either Shrink it, or Stable it.
To shrink a pet you will need a leash. These can be crafted, or purchased with vote tokens, and they are a reward for Taming Achievements. Double click the leash and target your pet to shrink it and place it in your pack where it will become a statue weighing 1 stone. Pets may not be shrunk while in combat.
Speak to an Animal Trainer npc, select Stable a Pet, and then click on the pet you wish to stable. For a small fee per day the animal will be kept safely in the stable. To retrieve your pet say 'Claim' followed by the name of the pet when you're at any stable. At any one time, a character may only have 3 pets stabled plus a bonus of 1 for each of the following.
Per 10 points of Animal Taming, Vetrenary, or Animal Lore over 100. (+1 at 110, +2 at 120, +3 at 130)
For having a total Animal Taming, Vetrenary, and Animal Lore of 240.
For having a total Animal Taming, Vetrenary, and Animal Lore of 300.