  Calculations of the gravitational self-force (GSF) in curved spacetime
require as input the metric perturbation in a sufficiently regular gauge. A
basic challenge in the program to compute the GSF for orbits around a Kerr
black hole is that the standard procedure for reconstructing the perturbation
is formulated in a class of radiation gauges, in which the particle singularity
is non-isotropic and extends away from the particle's location. Here we present
two practical schemes for calculating the GSF using a radiation-gauge
reconstructed metric as input. The schemes are based on a detailed analysis of
the local structure of the particle singularity in the radiation gauges. We
identify 3 types of radiation gauges: two containing a radial string-like
singularity emanating from the particle, either in one direction ("half-string"
gauges) or both directions ("full-string" gauges); and a third type containing
no strings but with a jump discontinuity across a surface intersecting the
particle. Based on a flat-space example, we argue that the standard
mode-by-mode reconstruction procedure yields the "regular half" of a
half-string solution, or (equivalently) either of the regular halves of a
no-string solution. For the half-string case, we formulate the GSF in a locally
deformed radiation gauge that removes the string singularity near the particle.
We derive a mode-sum formula for the GSF in this gauge, analogous to the
standard Lorenz-gauge formula but with modified regularization parameters. For
the no-string case, we formulate the GSF directly, without a local deformation,
and we derive a mode-sum formula that requires no correction to the parameters
but involves a certain averaging procedure. We explain the consistency of our
results with Gralla's invariance theorem, and discuss the correspondence
between our method and a related approach by Friedman et al.
