There is no breed of cat called the Minx cat although the term is widely found, and erroneously used, on the internet and usenet. Advertisements regularly appear for Minx Cats, adding further credence to this non-existent breed. The term is a common error made by people unfamiliar with the Manx cat, a tailless breed of cat originating from the Isle of Man, Great Britain.
Minx
Minx is sometimes colloquially used to indicate a miniature/dwarf form of Manx cat. It is not a separate breed.
Mynx
Mynx (with a 'y') is sometimes collooquially used to indicate a first generation cross between a Manx cat and Sphynx breed or a hybrid between the Manx breed and the wild Bobcat. Some bobcat hybrids are called Lynx e.g. Desert Lynx and American Lynx due to their resemblance to a lynx.