There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity.  And people often note a great thing about current proofs of FLT is their use of the modularity thesis which is just the opposite: arcane, and richly connected to a lot of results.

But have there been uses of FLT itself?  Beyond implying simple variants of itself, are there any more serious uses yet?

I notice the discussion in Fermat's Last Theorem and Computability Theory concludes that one purported use is not seriously using FLT.

Corollary 3.17 in this paper of Stefan Keil uses FLT for exponent 7 to show that if $E/\mathbb{Q}$ is an elliptic curve with a rational 7-torsion point $P$, and $E\rightarrow E'$ is the 7-isogeny with kernel $\langle P\rangle$, then $E'(\mathbb{Q})[7]=0$. There are of course lots of ways of proving this, but the paper does it by writing down a parametrisation of all elliptic curves over $\mathbb{Q}$ with 7-torsion and of their rational 7-isogenies, and then playing with parameters to get a contradiction to FLT.