  The self field of a charged particle has a component that diverges at the
particle. We use both coordinate and covariant approaches to compute an
expansion of this singular field for generic geodesic orbits in Schwarzschild
spacetime for scalar, electromagnetic and graviational cases. We check
agreement of both approaches and give, as an application, the calculation of
previously unknown regularisation parameters. In this so-called "mode-sum
regularization" approach, each mode of the field is finite, while their sum
diverges. The sum may be rendered finite and convergent by the subtraction of
"regularization parameters". Higher order parameters lead to faster convergence
in the mode-sum. As a second example application, we compute high order
expressions for the effective source approach to self-force calculations.
