Help Center

Answer verified

On classic version (browser version) go to Achievements in the blue task bar and then click Special Horses. Choose one of the categories and your objective is done.

Tip for new players: go try all the buttons in the task bar, lots of the objectives you get will have something to do with game functions you can find under these buttons.
And read the Breeder's Manual and the EC Manual, on the bottom of the Help Center page. Tons of info there too!