One of the most well-known curly-haired cat breeds is curly haired cats the Selkirk Rex, distinguished by its plush, curly coat and round face. The curls of the Selkirk Rex are not limited to a specific area but cascade over the entire body, creating a soft and tousled appearance. The coat is incredibly tactile, inviting people to run their fingers through the curls, reveling in the plush texture.