  We study the Iwasawa theory of $p$-primary Selmer groups of elliptic curves
$E$ over a number field $K$. Assume that $E$ has additive reduction at the
primes of $K$ above $p$. In this context, we prove that the Iwasawa invariants
satisfy an analogue of the Riemann--Hurwitz formula. This generalizes a result
of Hachimori and Matsuno. We apply our results to study rank stability
questions for elliptic curves in prime cyclic extensions of $\mathbb{Q}$. These
extensions are ordered by their absolute discriminant and we prove an
asymptotic lower bound for the density of extensions in which the Iwasawa
invariants as well as the rank of the elliptic curve is stable.
