  Linear perturbation is used to investigate the effect of gravitational
softening on the retrieved two-armed spiral eigenmodes of razor-thin stellar
discs.
  We explore four softening kernels with different degrees of gravity bias, and
with/without compact support (compact in the sense that they yield exactly
Newtonian forces outside the softening kernel). These kernels are applied to
two disc galaxy models with well-known unsoftened unstable modes. We illustrate
quantitatively the importance of a vanishing linear gravity bias to yield
accurate frequency estimates of the unstable modes. As such, Plummer softening,
while very popular amongst simulators, performs poorly in our tests.
  The best results, with excellent agreement between the softened and
unsoftened mode properties, are obtained with softening kernels that have a
reduced gravity bias, obtained by compensating for the sub-Newtonian forces at
small interparticle distances with slightly super-Newtonian forces at radii
near the softening length. We present examples of such kernels that, moreover,
are analytically simple and computationally cheap. Finally, these results light
the way to the construction of softening methods with even smaller gravity
bias, although at the price of increasingly complex kernels.
