Farrier or trimmer
A farrier works on the hoof capsule: trim, shoes, pulls, and the balance of the foot they can see and shape. A hoof-care trimmer is often doing the barefoot version of that job. Neither one is a veterinarian. If the horse is lame and you do not know why, the vet comes first or comes with them. Do not keep resetting a shoe on a mystery.
Equine dentist
Dental work is the mouth: sharp points, a bit seat, a wolf tooth, a mouth that is dropping feed. Rules about who may float teeth or sedate a horse are different from state to state. Ask the person, and if you are unsure, ask your veterinarian. A dentist does not replace a vet, and a vet visit for a cough is not a dental plan.
Teeth and feet get blamed for each other. A horse that is short-strided or sour under saddle might have a foot problem, a mouth problem, or something else. One visit does not rule the others out.
Mobile equine vet
Call the veterinarian for illness, wounds, colic signs, a horse that will not eat, a lameness you cannot explain, vaccines, and emergencies. A mobile practice comes to the farm, which matters when the horse should not get on a trailer. They can also tell you when the next person you need is the farrier, the dentist, or a clinic.
Horse hauler
A hauler is the right call when the horse has to go to a clinic, a new barn, or a show and you cannot take it. Ask what they drive, how horses are separated, whether they haul at night, and what insurance they carry. This site does not check policies. Get the proof yourself, and send a horse that will load.
If you are comparing categories on this directory, use the county pages. The person who shoes the horse, the person who looks at the mouth, the vet, and the hauler are four different trips.