I don't have an opinion of my own about Fuzzy Set Theory, but someone whom I respect, Saunders Mac Lane, seemed to think it wasn't a very fruitful development. (I freely acknowledge he could be opinionated at times, and he may have had his blind spots.) He mentions this example in his book Mathematics: Form and Function.
I invite others to explain to me what solutions to problems (or insightful points of view on problems) Fuzzy Set Theory has enabled, in the spirit of the proposed working definition of useful theory: something used to solve an existing problem that was posed before the theory was invented and that recordedly has withstood at least one previous attempt not using the new theory. Of course, fuzzy set theory may itself be a 'fuzzy' or at least broad term, in that it is cousin to other forms of set theory which do have their purpose, such as Heyting- or Boolean-valued set theory.