  The growing reality of gravitational wave astronomy is giving age-old
problems a new lease of life; one such problem is that of the self-force. A
charged or massive particle moving in a curved background space-time produces a
field that affects its motion, pushing it off its expected geodesic. This
self-field gives rise to a so-called self-force acting on the particle. In
modelling this motion, the self-force approach uses a perturbative expansion in
the mass ratio. One of the most interesting sources of gravitational waves are
extreme mass ratio inspirals - systems perfectly suited to self-force
modelling. One of the key problems within the self-force model is the
divergence of the field at the particle. To resolve this, the field is split
into a singular component and a smooth regular field. This regular-singular
split, introduced by Detweiler and Whiting, is used in most modern self-force
calculations. In this thesis, we derive high-order expansions of the
Detweiler-Whiting singular field, and use these to push the boundaries on
current precision limits of self-force calculations. Within the mode-sum
scheme, we give over 14 previously unknown regularisation parameters, almost
doubling the current regularisation parameter database. We also produce smooth
effective sources to high order, and propose an application of the higher terms
to improve accuracy in the m-mode scheme. Finally, we investigate the status of
the cosmic censorship conjecture and the role that the self-force plays. To
this end, we give regularisation parameters for non-geodesic motion. We also
show the necessity of our results in the exciting area of second order
self-force calculations, which benefit significantly from high-order coordinate
expansions of the singular field. We calculate several parameters that these
schemes require, and highlight the further advancements possible from the
results of this thesis.
