  To model the radiative evolution of extreme mass-ratio binary inspirals (a
key target of the LISA mission), the community needs efficient methods for
computation of the gravitational self-force (SF) on the Kerr spacetime. Here we
further develop a practical `$m$-mode regularization' scheme for SF
calculations, and give details of a first implementation. The key steps in the
method are (i) removal of a singular part of the perturbation field with a
suitable `puncture' to leave a sufficiently regular residual within a finite
worldtube surrounding the particle's worldline, (ii) decomposition in azimuthal
($m$-)modes, (iii) numerical evolution of the $m$-modes in 2+1D with a finite
difference scheme, and (iv) reconstruction of the SF from the mode sum. The
method relies on a judicious choice of puncture, based on the
Detweiler--Whiting decomposition. We give a working definition for the `order'
of the puncture, and show how it determines the convergence rate of the
$m$-mode sum. The dissipative piece of the SF displays an exponentially
convergent mode sum, while the $m$-mode sum for the conservative piece
converges with a power law. In the latter case the individual modal
contributions fall off at large $m$ as $m^{-n}$ for even $n$ and as $m^{-n+1}$
for odd $n$, where $n$ is the puncture order. We describe an $m$-mode
implementation with a 4th-order puncture to compute the scalar-field SF along
circular geodesics on Schwarzschild. In a forthcoming companion paper we extend
the calculation to the Kerr spacetime.
