  Firstly, we see that the bases of the miniversal deformations of isolated
$\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In
particular, we classify the analytic germs of embedding dimension $\leq 2$
which are the bases of the miniversal deformations of isolated
$\mathbb{Q}$-Gorenstein toric singularities.
  Secondly, we show that the deformation spaces of isolated Gorenstein toric
$3$-fold singularities appear, in a weak sense, as singularities of the
K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a
consequence, we prove that the number of local branches of the K-moduli stack
of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano
varieties is unbounded in each dimension $\geq 3$.
