  We study conditions under which the germinal groupoid associated to a minimal
equicontinuous action of a countable group on a Cantor set has non-Hausdorff
topology. We develop a new criterion, which serves as an obstruction for the
\'etale topology on the groupoid to be non-Hausdorff. We use this and other
criteria to study the topology of germinal groupoids for a few classes of
actions. In particular, we give examples of families of contracting and
non-contracting self-similar groups, which are amenable and whose actions have
associated germinal groupoids with non-Hausdorff topology.
