  Recently, we proposed a method for calculating the ``radiation reaction''
self-force exerted on a charged particle moving in a strong field orbit in a
black hole spacetime. In this approach, one first calculates the contribution
to the ``tail'' part of the self force due to each multipole mode of the
particle's self field. A certain analytic procedure is then applied to
regularize the (otherwise divergent) sum over modes. This involves the
derivation of certain regularization parameters using local analysis of the
(retarded) Green's function. In the present paper we present a detailed
formulation of this mode-sum regularization scheme for a scalar charge on a
class of static spherically-symmetric backgrounds (including, e.g., the
Schwarzschild, Reissner-Nordstr\"{o}m, and Schwarzschild-de Sitter spacetimes).
We fully implement the regularization scheme for an arbitrary radial trajectory
(not necessarily geodesic) by explicitly calculating all necessary
regularization parameters in this case.
