  Fix two distinct odd primes $p$ and $q$. We study "$p\ne q$" Iwasawa theory
in two different settings. Let $K$ be an imaginary quadratic field of class
number 1 such that both $p$ and $q$ split in $K$. We show that under
appropriate hypotheses, the $p$-part of the ideal class groups is bounded over
finite subextensions of an anticyclotomic $\mathbb{Z}_q$-extension of $K$. Let
$F$ be a number field and let $A_{/F}$ be an abelian variety with
$A[p]\subseteq A(F)$. We give sufficient conditions for the $p$-part of the
fine Selmer groups of $A$ over finite subextensions of a
$\mathbb{Z}_q$-extension of $F$ to stabilize.
