In France, the Government actively supports Equestrianism and it is a widely popular activity. It is well organised with a professional structure of trainers, breeders and riders. The entire country is divided into French Equestrian Federation (FFE) regions and equestrian sport is organised within and between regions. Riders are expected to progress through a licencing system, such that before one enters a show a minimum level of competence much be achieved. Horses are dispersed across the country but Normandy is considered to be the main centre, with a high number of stud farms. There is fantastic out riding throughout France as the nation is criss crossed with accessible trails maintained by the local communes. Endurance riding is probably the best French discipline. It is a lovely country for riding vacations.
Have a look at www.ffe.com , which will translate into English for you, as your starting point. Look for lists of stables and trainers by region.
As a personal aside, I can think of several top notch French Show Jumpers and Eventers but no Dressage star springs to mind. The national breed, the Selle Francais, is mainly focused on jumping ability.