Maseches Shabbos, Chapter 5, Mishnah 1. This chapter takes up the question of what an animal may go out with on Shabbos - if one may call it wearing - and what it may not go out with. The verse in Shemos 23 says: "So that your ox and your donkey may rest," and from here we learn that a person is obligated in the resting of his animal, to ensure that his animal too rests on Shabbos.
Defining the animal's rest:
The rest required of an animal is not identical to the rest of a person. The intent is that it should not perform labor for a person and should not carry a burden for him. But something that serves the animal itself - for its safety, for keeping it under control, so that it should not run wild, and even an ornament, provided it is an ordinary and accepted ornament - is considered subordinate to the animal. Such items are regarded as part of the animal's own needs, and they involve neither carrying nor any problem of labor on Shabbos. The question is how to define this.
And so the Mishnah opens: "Bameh behemah yotzeah u'vameh einah yotzeah" - with what may an animal go out into a public domain on Shabbos, and with what may it not go out. And the Mishnah lists examples:
"Yotzei hagamal b'afsar" - a camel may go out with a halter, a rope that was tied by a chain to the side of its face, and by means of it the animal was led.
"U'nakah b'chatam" - a nakah is a female camel, and the chatam is a ring that was placed in its nose. They would pierce the nose of the nakah and insert there a metal fixture shaped like a ring, in order to keep the animal under control, since a nakah needs greater control than the male.
"V'lubdekis b'promvia" - the lubdekis is a type of donkey that came from the country of Luv (Libya). This donkey was unusually strong and required far more precautions to keep it under control than other donkeys, and therefore it goes out with a promvia, a bit made of iron.
"V'sus b'shir" - a horse may go out with a shir, a fixture placed around its neck.
"V'chol ba'alei hashir yotz'im b'shir v'nimshachin b'shir" - all types of animals that are customarily treated this way have such a fixture around the neck: a sort of chain, or anything placed around their neck to protect them or to take hold of them. All of them may go out with the shir, because this is an ordinary thing, and one may even lead them by means of it.
Sprinkling and immersing the shir:
A metal shir that became tamei with the tumah of a corpse, for example by coming into contact with a corpse, and requires sprinkling upon it of the waters of the ashes of the red heifer - one may sprinkle upon it while it is still on the animal. And afterward one may also immerse it in a mikvah while it is still on the animal, since it is loose enough, and therefore one may bring the animal into the water and immerse the shir in the mikvah.
And even though the ordinary halachah is that vessels designated specifically for animals are not susceptible to tumah, the shir and vessels like it are susceptible to tumah and require immersion in a mikvah. The reason is that since a person uses it to lead the animal, it is like a vessel designated for a person's use, and therefore it is susceptible to tumah and requires purification - both through sprinkling the ashes upon it and through bringing it into a mikvah. And since the shir is loose, the waters of the mikvah can enter between the animal's neck and the shir, and therefore it is able to become tahor.
In summary: In this Mishnah we learned the principle of the resting of one's animal, derived from the verse "So that your ox and your donkey may rest," and the distinction between a burden or labor for a person's needs and something that is subordinate to the animal and serves to keep it under control and to meet its needs. The Mishnah listed the items with which animals go out on Shabbos: the camel with a halter, the nakah with a chatam, the lubdekis with a promvia, and the horse and all animals that wear a shir with a shir. And finally we examined the laws of tumah and taharah regarding the shir, which is susceptible to tumah because a person uses it, and whose looseness makes it possible to sprinkle upon it and immerse it while it is still on the animal.