  We consider the space of $C^1$-diffeomorphims equipped with the
$C^1$-topology on a three dimensional closed manifold. It is known that there
are open sets in which $C^1$-generic diffeomorphisms display uncountably many
chain recurrences classes, while only countably many of them may contain
periodic orbits. The classes without periodic orbits, called aperiodic classes,
are the main subject of this paper. The aim of the paper is to show that
aperiodic classes of $C^1$-generic diffeomorphisms can exhibit a variety of
topological properties. More specifically, there are $C^1$-generic
diffeomorphisms with (1) minimal expansive aperiodic classes, (2) minimal but
non-uniquely ergodic aperiodic classes, (3) transitive but non-minimal
aperiodic classes, (4) non-transitive, uniquely ergodic aperiodic classes.
