  We calculate the singular field of an accelerated point particle (scalar
charge, electric charge or small gravitating mass) moving on an accelerated
(non-geodesic) trajectory in a generic background spacetime. Using a mode-sum
regularization scheme, we obtain explicit expressions for the self-force
regularization parameters. In the electromagnetic and gravitational case, we
use a Lorenz gauge. This work extends the work of Barack and Ori [1] who
demonstrated that the regularization parameters for a point particle in
geodesic motion in a Schwarzschild spacetime can be described solely by the
leading and subleading terms in the mode-sum (commonly known as the $A$ and $B$
terms) and that all terms of higher order in $\ell$ vanish upon summation
(later they showed the same behavior for geodesic motion in Kerr [2], [3]). We
demonstrate that these properties are universal to point particles moving
through any smooth spacetime along arbitrary (accelerated) trajectories. Our
renormalization scheme is based on, but not identical to, the Quinn-Wald
axioms. As we develop our approach, we review and extend work showing that that
different definitions of the singular field used in the literature are
equivalent to our approach. Because our approach does not assume geodesic
motion of the perturbing particle, we are able use our mode-sum formalism to
explicitly recover a well-known result: The self-force on static scalar charges
near a Schwarzschild black hole vanishes.
