  For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set
$K=\mathbb{Q}(E[p])$ to denote the torsion field generated by
$E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group
$\operatorname{Cl}_K$ is a module over $\operatorname{Gal}(K/\mathbb{Q})$.
Given a fixed odd prime number $p$, we study the average non-vanishing of
certain Galois stable quotients of the mod-$p$ class group
$\operatorname{Cl}_K/p\operatorname{Cl}_K$. Here, $E$ varies over rational
elliptic curves, ordered according to \emph{height}. Our results are
conditional and rely on predictions made by Delaunay and Poonen-Rains for the
statistical variation of the $p$-primary parts of Tate-Shafarevich groups of
elliptic curves. We also prove results in the case when the elliptic curve
$E_{/\mathbb{Q}}$ is fixed and the prime $p$ is allowed to vary.
