  We compare the Pontryagin duals of fine Selmer groups of two congruent
$p$-adic Galois representations over admissible pro-$p$, $p$-adic Lie
extensions $K_\infty$ of number fields $K$. We prove that in several natural
settings the $\pi$-primary submodules of the Pontryagin duals are
pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer
groups are not equal, then we can still prove inequalities between the
$\mu$-invariants. In the special case of a $\mathbb{Z}_p$-extension
$K_\infty/K$, we also compare the Iwasawa $\lambda$-invariants of the fine
Selmer groups, even in situations where the $\mu$-invariants are non-zero.
Finally, we prove similar results for certain abelian non-$p$-extensions.
