Dogs, cats, birds, reptiles, fish, etc. are all regarded as property. All the stuff that goes along with them (bedding, tanks/cages, toys, etc.) is awarded to whoever is keeping the pet. It's just like the notion that children's furniture goes with the children. Unfortunately, as far as financial value is concerned regular housepets have virtually no monetary value. While some pets may have unusual expenses or require a lot of of support equipment (think: fish), the pets themselves have no re-sale value. In fact, unless there is some special financial issue courts will not address pet issues. They normally will not hear argument about who should get the dog or cat, etc. There are no pattern court orders that set "custody" or "visitation" with pets. As a retired court commissioner famously said in my jurisdiction, "I don't do dogs."
Pets that are "show quality" have a monetary value that can be assessed. The pet needs to have "papers," documentation of bloodlines and be registered with a governing authority (such as AKC) to establish that the animal is of show quality. It doesn't matter if the pet is being actively shown or is used only for breeding. It's the fact that the pet has economic value in an open market and can be assigned a fair market value. The fact that you may have paid a lot of money for a pet or purchased a pet from a breeder doesn't make a pet "show quality." Many breeders sell purebred animals that are merely "pet quality." If you purchased your animal from a breeder, check with the breeder about what level of quality is assigned to your pet. "Show quality" animals' value is to be taken into consideration when dividing assets in a dissolution. However, determining who gets the pet or if the other party gets "visitation" is not an issue the court will look at.
Horses, cattle, sheep and other livestock tend to have a monetary value that can be objectively assessed. Thus they are part of the property division. The purpose of the livestock doesn't matter (although it may change how the animal's value is calculated). An actively racing thoroughbred is as much property as a bull kept for breeding. An animal retired from its activity (i.e., racing, breeding, showing) and is just being allowed to live out the rest of its natural life may or may not retain its economic value. An expert in that type of animal can better assess the animal's value. Livestock also have significantly more expensive supplies. Trailers are a common item to include in property division calculations. In cases involving livestock, it's best to have an expert establish a value for the animals.
When you can't keep the pet
Please, please, please if you have to give up a pet seek out a no-kill shelter. Many families dump their dogs and cats (and sometimes other animals) when they have to move from the family home. Animals don't understand divorce. Moving from a house with a yard to a small apartment or from a hot climate to a cold one is like taking away an animal's entire world. While the financial devastation that goes along with divorce dictates the choices left for the parties, a little extra time to make sure that any pets will continue to have a safe loving home is so important. No matter how much you and your soon-to-be ex hate each other, please remember that your children and your animals have no way to tell the court what should happen to them.