Yes, American Curl cats are healthy because cats of this breed don’t inherit genetic illnesses (as at 1992). I think I have the best book that you can buy to answer the question in the title. The book that I’m referring to is Medical, Genetic and Behavioral Aspects of Purebred Cats, Edited by Ross D Clark DVM. On pages 35 and 36 he discusses this cat breed and under ‘Recognised Medical Problems’ the author writes: “At this time, there is no evidence of any specific disorders common to the American Curl”. Although the book was published some time ago in 1992, I find this reassuring nonetheless. Of course, the cat’s owner has to maintain the health of a what is essentially a healthy cat.
The question in the title is asking obliquely whether the American Curl is a breed with genetically inherited defects. Often purebred cats do inherit certain conditions because that is the nature of line-breeding. If you selectively breed you can reveal genetic mutations which are unhelpful. The good news is that the American Curl is considered healthy. The mutation that causes the ears (the cartilage in the pinnae) to curl back does not carry any associated side effects. Clearly, the mutated gene affects the cartilage in the ear flap. Sometimes when the cartilage is affected in this way other parts of the body are also affected but that is not the case in this instance, as I understand it. It is a dominant gene which mutates which results in 50% curly-eared kittens in litters resulting from crosses between cats with curled ears and cats with normal ears. The cat is unconnected to the better-known Scottish Fold.
The American Curl cannot be described as a popular cat breed. Here are 2 American Curls: