  We investigate the impact of topology on the existence of impurity subgap
states in a time-reversal-invariant superconductor with an extended $s$-wave
pairing and strong spin-orbit coupling. By simply tuning the chemical potential
we access three distinct phases: topologically trivial $s$-wave, topologically
non-trivial $s_\pm$-wave, and nodal superconducting phase. For a single
potential impurity we find subgap impurity bound states in the topological
phase, but notably no subgap states in the trivial phase. These subgap impurity
states have always finite energies for any strength of the potential scattering
and subsequently, the superconducting gap in the topological $s_\pm$-wave phase
survives but is attenuated in the presence of finite disorder. By creating
islands of potential impurities we smoothly connect the single impurity results
to topological edge states of impurity island. On the other hand, magnetic
impurities lead to the formation of Yu-Shiba-Rusinov states in both the trivial
and topological phases, which even reach zero energy at certain scattering
strengths. We thus propose that potential impurities can be a very valuable
tool to detect time-reversal-invariant topological superconductivity.
