  We consider several (related) notions of geometric regularity in the context
of limit sets of geometrically finite Kleinian groups and associated
Patterson-Sullivan measures. We begin by computing the upper and lower
regularity dimensions of the Patterson-Sullivan measure, which involves
controlling the relative measure of concentric balls. We then compute the
Assouad and lower dimensions of the limit set, which involves controlling local
doubling properties. Unlike the Hausdorff, packing, and box-counting
dimensions, we show that the Assouad and lower dimensions are not necessarily
given by the Poincar\'e exponent.
