The answer is: it depends!
If you are looking at hunter divisions, you are not allowed to enter the under saddle class if the horse has not completed at least one course over fences in that division. This means that someone, not necessarily you, must take the horse in a minimum of one class in that division before you can ride on the flat. Because hunters are judged on the horse, the rider for the over fences does not need to be the same as the rider for the flat, provided both riders meet the criteria for that division. The Amateur Owner division is a special exception because riders who have multiple horses in the division may use another amateur who is not an owner to show any extra horses on the flat.
ARTICLE G316 OTHER HUNTER CLASSES
1. Hunter under saddle: to be shown at a walk, trot and canter both ways of the ring; at least eight horses, at the judge's discretion, if available, may be required to gallop one way in the ring but never more than eight at one time (pre-green and green hunters not to gallop). Light contact with the horse's mouth is permissible. In order to maintain awards, horses must compete and complete the course in at least one over fences class in their respective division. Hunter under saddle classes must never be the first class of a division.
If you are looking at the equitation divisions, you are allowed to enter a flat class without entering the jumping class in the same division.
If you are looking at doing flat classes only, keep in mind that equitation divisions generally restrict cross-entry into other equitation divisions because they are divided by age or experience. You will be restricted to a single equitation on the flat class per horse show.
If you would like to enter multiple flat classes at a horse show without doing any jumping, you will have to either look for a show that offers a hack division or have another rider take your horse over fences in order to qualify for a hunter under saddle class.