  Let $p$ be an odd prime. Associated to a pair $(E, \mathcal{F}_\infty)$
consisting of a rational elliptic curve $E$ and a $p$-adic Lie extension
$\mathcal{F}_\infty$ of $\mathbb{Q}$, is the $p$-primary Selmer group
$Sel_{p^\infty}(E/\mathcal{F}_\infty)$ of $E$ over $\mathcal{F}_\infty$. In
this paper, we study the arithmetic statistics for the algebraic structure of
this Selmer group. The results provide insights into the asymptotics for the
growth of Mordell--Weil ranks of elliptic curves in noncommutative towers.
