Well, there are a number of necroids of myth and legend. In general, these can be divided into two categories: the remains of a dead creature’s mind (animoids), or the remains of a dead creature’s body (cadaveroids). Some creatures can be both.
Ghosts would belong to the former, as they are usually described as non-corporeal remnants of dead ponies. They can be benign but often they are not, either trying to cause trouble for ponies (usually to drive them away), scare ponies or even wish to cause ponies harm. Though it is worth noting the necroid ghosts are not to be confused with real corporeal beings somewhat mistakenly referred to as ghosts, such as the crystal ghosts which are indigenous to the Crystal Empire, albeit rare.
Zombie ponies are another common myth, these are almost always described as slow, lurching reanimated corpses in varying states of decay, with no capacity for thought. Sometimes they are pawns of some evil force but can also just be mindlessly trying to eat brains (why it’s brains specifically, I have never figured out, carnivores would typically want to eat muscle tissue).
Skeletons are similar to zombie ponies in that they are often described as reanimated to serve a malevolent force, but unlike zombie ponies they are almost always in a state of decay where only the skeleton remains, and they do not try to eat brains in any of the legends I know.
Zombie ponies and skeletons are both very common enemies in epic tales and role-playing games. In Black and White: The Ethics and Symbolism of Myths, Pensive Prose posits that because they are already dead, thus going against the natural order, and because they lack sentience, this makes them the perfect enemy. You never have to feel bad about fighting them or question the ethics of doing so, thus it is always clear who is the hero in the story.
An example of a creature which falls into both categories is the Balehorn, a unicorn who uses forbidden magic to sustain the mind and body after death. While they often appear to look like zombie ponies or skeletons, the difference is that they became undead willingly and maintain their mind and magical ability and are thus far more dangerous. Often these are the main antagonists responsible for reanimating zombies and skeletons.
Note that I’ve only covered a few, there really are many more necroid creatures in fiction. Because they’re such convenient enemies, as detailed previously, many writers use them as antagonists and try to come up with new ones to be less derivative, leading to a myriad of them being thought up.