When I think of "Fuzzy logic" the first thing that comes to mind is engineering, where is it used in control systems, and information processing. For example, "fuzzy logic" is used in control systems for washing machines. I think it was a hot topic in 1980s and 1990s, and one article from that time is "Designing with Fuzzy Logic" by Kevin Self from the IEEE Spectrum magazine in 1990.
My opinion as a logician is that "fuzzy logic" per se has not developed significant interest in mathematical logic for the study of foundations of mathematics. I have not seen any examples of "deep" applications of fuzzy logic to classical foundational issues. One reason why it will be difficult to develop deep foundational implications of fuzzy logic is that much work in foundations is motivated by a few standard structures (ignoring any philosophical objections): the natural numbers, the real numbers, the cumulative hierarchy of sets. These are not ordinarily conceived as "fuzzy" structures.
As a personal aside, I would not use the term "fuzzy logic" to refer to multivalued logic in general, and I think it is somewhat ahistorical to refer to work of Łukasiewicz as "fuzzy logic". However, others do use "fuzzy logic" more broadly, and there is a book by Hájek entitled "Metamathematics of Fuzzy Logic", so my opinion is far from universal.
None of this means there is a lack of value in fuzzy logic, of course. It is obviously of great interest for information processing, control theory, and engineering, where it may indeed have deep aspects.
The introduction to the Stanford Encyclopedia of Philosophy article on fuzzy logic also remarks on some of these issues. In that terminology, I am apparently speaking about "fuzzy logic in the broad sense".