Any accessory worn by an animal for purposes of shmira is allowed to be worn into the reshus horabim. Anything else is a masui. For example: Accessories for excessive shmira. Adornments. Clothing (excluding warming saddles for donkeys only). (ShA 1, 7 – MB 27)
Your animal may wear a collar – it is for shmira, as you can guide it by pulling the collar. The only condition is that the collar in question is normally used for the type of animal that is wearing it. (ShA 5 – MB 12, 19)
The leash may be wrapped around the animal’s neck. You might use it to pull your animal. (ShA 1 – MB 21)
When you hold the leash in the reshus horabim, make sure that the end of the leash does not extend beyond where you are holding it more than a tefach. (ShA 16)
The leash must be short. If it is so long that it sags within a tefach above the ground it is ossur to “carry” it outside. You can wrap the excess around the animal's neck. (ShA 16 – MB 51)
Chazal decreed that it is ossur to ride animals on Shabbos. This includes even leaning on the side of an animal, and even riding in a wagon hitched to an animal. (ShA 18)
You are allowed to lean on the side of something that covers the side of the animal. (ShA 18 – MB 63)
We pasken that it is a din d’oraisa to save an animal from suffering. (MB 69)
You are allowed to instruct a gentile to do a melacha which will alleviate a suffering animal. Example: Milking a mother animal. (You may not drink the milk until after Shabbos. ) It would be commendable to instruct the gentile to squirt the milk directly into the food – it reduces the act to “shvus d’shvus” . (ShA 20 – MB 73)