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When it comes to shoeing, dressage, hunter jumpers, and event horses all have different needs and ways of going.
The basics: Trimming and shoeing the green horse.
All horses should be trimmed the same. A well-balanced foot with smooth edges is an absolute necessity for any horse, but especially an active one. Many lameness issues arise from or are enhanced by the trim.
A horse that is not working much doesn't usually need shoes. In fact, shoes aren't necessary unless the horse is not able to do its job comfortably without them or there is some advantage that will make him work or move better.
The level of training corresponds to what level of shoeing is needed. A horse should have basic training using basic shoes. This allows the horse to learn naturally.
More advanced training often requires more advanced shoeing to aid them in their job. That doesn't speak to quality, but the two should always be the same.
Dressage
Dressage horses do a lot of upward and downward transitions and lateral movements. Power from hind quarters requires balance of traction, flotation, and support of limb to aid movement in the ring.
Front hoof balance for an upper level dressage horse is critical because the concussion from an extension landing can cause any imbalance to show as lameness.
Dressage riders can tell when their horses are not moving perfectly, and a farrier should watch the horse work to find out how to balance the horse optimally.
When shoeing dressage horses, I like to use a wide-webbed relatively heavy steel shoe up front; often I use a perimeter fit (meaning shoe is fit to the edges of the hoof and not set back off the toe).
The rear hooves are shod with a slightly more narrow webbed steel shoe for traction that will aid impulsion.
Many dressage riders say they like a trailer on the outside branch or at least a wider webbed lateral (outside) branch for the hind shoes.
Hunter Jumpers
Hunter jumpers are more on the forehand than dressage horses. Horses move in a smoother style with less knee and hock movement or "action." The weight of the shoes is important because a heavy shoe will cause more action.
Most equitation horses use aluminum shoes on the front and light steel behind. Properly applied break-over (shoe set back off the toe) of the front and hind can enhance the movement.
Lower-level jumpers often have some sort of permanent low level traction applied to the shoes and higher jumpers often have "drilled and tapped" holes which the rider can use to add, remove, or change traction as needed.
Event Horses
Event horses need solid shoes which are well designed for balance of weight, durability, traction, support and staying on in extreme conditions.
As a horse that does basically all of the above with some real galloping thrown in, they have to stay in the middle of how the dressage and hunter jumpers are "optimized".
Many event riders prefer a "concave" shoe that cuts into the ground more than the flat shoe of the other disciplines. Shoeing for eventers is often highly customized.