1. The White Chocobo
Supposedly, with a white chocobo (and the Underwater materia) you can travel underwater without the need for a submarine. This is the most rampant chocobo rumor I've ever seen.
2. The Red Chocobo
This chocobo supposedly flies in the air, just like the Highwind. I don't know the breeding process, though
3. The Wooden Chocobo
This rumor is real funny. The breeding process supposedly involves you stealing the underwear of Yuffie's father. Then you should run around in one of the forests for a bit, and you'll end up fighting a Wooden Chocobo which you can capture, or something to that effect.
4. Teoih's Chocobo
This rumor sounds reasonable at least. You can get Teoih's chocobo by having him join your party.