According to the [2007 survey of Harmon, Diamond, and Taylor][1], in 1985 "Fermat’s Last Theorem was still not known to be true for an infinite set of prime exponents".  [Note: I found this paper using ChatGPT.  Normally I would have found it using a simple Google search, but with the advent of AI searching that way has become much worse!]

I suspect that the later FLT developments pre-Wiles also did not establish infinitely many prime exponent cases, but those more familiar with the Taniyama-Shimura connection can weigh in.


  [1]: https://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf