  We study six-dimensional rotating black holes with bumpy horizons: these are
topologically spherical, but the sizes of symmetric cycles on the horizon vary
non-monotonically with the polar angle. We construct them numerically for the
first three bumpy families, and follow them in solution space until they
approach critical solutions with localized singularities on the horizon. We
find strong evidence of the conical structures that have been conjectured to
mediate the transitions to black rings, to black Saturns, and to a novel class
of bumpy black rings. For a different, recently identified class of bumpy black
holes, we find evidence that this family ends in solutions with a localized
singularity that exhibits apparently universal properties, and which does not
seem to allow for transitions to any known class of black holes.
