There are at least four main types of competitive pigeon sport:
Pigeon racing Tumbling Highflying Tippler (Endurance)
Though not quite a sport, fancy breeds of pigeons are also bred to standards and judged in a competitive fashion. Levi in his book The Pigeon describes all aspects of pigeon keeping. For exhibition purposes sport pigeons are sometimes grouped as Flying/Sporting Pigeons.
References
See also Birmingham Roller List of pigeon breeds