  We present an exhaustive numerical investigation of the optical caustics in
gravitational lensing by a spinning black hole for an observer at infinity.
Besides the primary caustic, we examine higher order caustics, formed by
photons performing one or several loops around the black hole. Our
investigation covers the whole parameter space, including the black hole spin,
its inclination with respect to the line of sight, the source distance, and the
caustic order. By comparing our results with the available analytical
approximations, we find perfect agreement in their respective domains of
validity. We then prove that all caustics maintain their shape (a tube with
astroidal cross-section) in the entire parameter space without suffering any
transitions to different caustic shapes. For nearly extremal spin, however,
higher order caustics grow so large that their cross-sections at fixed radii
wind several times around the black hole. As a consequence, for each caustic
order, the number of images ranges from 2 to 2(n+1), where n is the number of
loops spanned by the caustic. As for the critical curves, we note that for high
values of the spin they develop a small dip on the side corresponding to
prograde orbits.
