  Let $K_\infty/K$ be a uniform $p$-adic Lie extension. We compare several
arithmetic invariants of Iwasawa modules of ideal class groups on the one side
and fine Selmer groups of abelian varieties on the other side.
  If $K_\infty$ contains sufficiently many $p$-power torsion points of $A$,
then we can compare the ranks and the Iwasawa $\mu$-invariants of these modules
over the Iwasawa algebra. In several special cases (e.g. multiple
$\mathbb{Z}_p$-extensions), we can also prove relations between suitable
generalisations of the Iwasawa $\lambda$-invariant of the two types of Iwasawa
modules.
  In the literature, different kinds of Iwasawa $\lambda$-invariants have been
introduced for ideal class groups and Selmer groups. We define analogues of
both concepts for fine Selmer groups and compare the resulting invariants.
  In order to obtain some of our main results, we prove new asymptotic formulas
for the growth of ideal class groups and fine Selmer groups in multiple
$\mathbb{Z}_p$-extensions.
