This is a full list of all the breeds in the game as of October 29th, 2016.
I have changed up the color key a little, simply because the majority of the breeds are Personal Breeds. Now public breeds are color coded to make it easier to see just those on the list. Private breeds are all listed in black.
Legend:
Special Event Breed (Holiday, gift, and event breeds)
Sponsor Reward Breed
Catchable Breed
Sterile At Birth/Unable to Breed
Unknown Type
Personal Breed
Normal Public Breed