  In a previous paper, we computed expressions for the Detweiler-Whiting
singular field of point scalar, electromagnetic and gravitational charges
following a geodesic of the Schwarzschild spacetime. We now extend this to the
case of equatorial orbits in Kerr spacetime, using coordinate and covariant
approaches to compute expansions of the singular field in scalar,
electromagnetic and gravitational cases. As an application, we give the
calculation of previously unknown mode-sum regularization parameters. We also
propose a new application of high-order approximations to the singular field,
showing how they may be used to compute $m$-mode regularization parameters for
use in the $m$-mode effective source approach to self-force calculations.
