The Pet Semetary
There is a long tradition of animals being buried along with their human masters. Many plains tribes people would bury the dead with their horses, and the Egyptians would mummify the cats of the ruling classes. Rather than being simply buried in the ground, burnt, or otherwise disposed of at the end of their lives, the animal dead would still have a place in a setting where necromancy existed.
The spell to create a cadaverous companion is little different from the typical animate dead spell. The corpse must be cleaned and prepared for reanimation, anointed with oils and aromatic herbs, part for the magic, and part to conceal the stench of rotting flesh. Once cleaned, the body is laid on on a ceremonial altar and it is painted with blood runes and a large stone, smoothed and polished, is pushed into the throat. The less pleasant alternative is that the entire lower jawbone is removed. This reduces the likelihood of an animated companion breaking loose and decided to start gorging itself on flesh.The process typically takes a week, allowing for a period of drying and preserving.
The undead created by the spell are different than typical undead. They are much more docile, and easily controlled. The spell embeds a certain servitude into the reanimated corpse. This is important as most animal handlers already have a hard enough time caring for live animals that can hurt them, dealing with immune to pain, unsleeping, hungry for flesh versions of their living selves just makes it that much worse.
The King's Horse
The finest horse in the kingdom, this animal lived a life of great comfort, rivaling even the wealthy of the kingdom. The King, remembering riding into fateful battle atop said horse was not willing to let the beast go when it finally succumbed to age. Instead, it was preserved by the finest sarcomancers. It was oiled and polished until it's dead flesh gleamed, and then they rose the creature to stand again. No longer interested in mating with the mares, or feeding in the pasture, the horse stands silent, waiting. There are few things that stir it from it's silence. The King still rides the fell animal when on campaign, and to the foes of the kingdom, there are few sights as fear inspiring as the King and his bannermen, riding up clad in heavy armor riding on tireless, bloodless, fearless, monsters.
The Bloodless Hounds
Diseases are common, and it can make keeping a pack of hunting or war dogs difficult. The Bloodless Hounds are large canines bred for tracking, fighting, and killing. The spell renders them fearless and bloodless, and their taste for flesh makes them even better trackers than in life. Handlers of Bloodless Hounds tend to keep holy symbols on hand to restrain a pack that breaks free of their control. Unlike the normal version of the spell, the animals are left with their maws intact, otherwise their inclination to hunt is greatly diminished.
The Deathless Beasts
A menagie is an expensive investment for the ultra wealthy and powerful, and sometimes the expensive animals have a bad habit of dying. A Deathless Beast is an exotic, likely imported, animal such as the African megafauna giraffes, elephants, and such. Once preserved the animals have no purpose other than to wander around their enclosures of cages. These undead are kept confined as they are much more dangerous compared to more mundane livestock or companion animals. Birds can be animated, but the expense is much greater, the chance of failure is higher, and side from using adhesives, their feathers are going to fall off at a fairly quick pace.
Cadaverous Companions are a sure sign of a necromancy-friendly society, and as such it won't be out of sorts to see reanimated human corpses given similar treatment and being used in manual labor roles. Zombies under the command of a corpse drover can pull a wagon or plow just as well as livestock, and without the matter of providing large amounts of forage for them to turn into equally large amounts of excrement.
The largest weakness of Cadaverous Companions is that they are very susceptible to clerical turning abilities, and they are easily destroyed in this manner. Being undead, they are also vulnerable to being taken over by greater undead and being turned against their handlers. This is a less common issue, as most places with friendly relationships with the undead are likely already associated with things like vampires and liches.