  Given a prime number $p$, we study topological analogues of Iwasawa
invariants associated to $\mathbb{Z}_p$-covers of the $3$-sphere that are
branched along a link. We prove explicit criteria to detect these Iwasawa
invariants, and apply them to the study of links consisting of $2$ component
knots. Fixing the prime $p$, we prove statistical results for the average
behaviour of $p$-primary Iwasawa invariants for $2$-bridge links that are in
Schubert normal form. Our main result, which is entirely unconditional, shows
that the density of $2$-bridge links for which the $\mu$-invariant vanishes,
and the $\lambda$-invariant is equal to $1$, is $(1-\frac{1}{p})$. We also
conjecture that the density of $2$-bridge links for which the $\mu$-invariant
vanishes is $1$, and this is significantly backed by computational evidence.
Our results are proven in a topological setting, yet have arithmetic
significance, as we set out new directions in arithmetic statistics and
arithmetic topology.
