  The set of all metrics that can be placed on a given manifold defines an
infinite-dimensional `superspace' that can itself be imbued with the structure
of a Riemannian manifold. Geodesic distances between points on Met$(M)$ measure
how close two different metrics over $M$ are to one another. Restricting our
attention to only those metrics that describe physical black holes, these
distances may therefore be thought of as measuring the level of geometric
similarity between different black hole structures. This allows for a
systematic quantification of the extent to which a black hole, possibly arising
as an exact solution to a theory of gravity extending general relativity in
some way, might be `non-Kerr'. In this paper, a detailed construction of a
superspace for stationary black holes with an arbitrary number of hairs is
carried out. As an example application, we are able to strengthen a recent
claim made by Konoplya and Zhidenko about which deviation parameters describing
a hypothetical, non-Schwarzschild black hole are likely to be most relevant for
astrophysical observables.
