  Band structures of topological insulators are characterized by non-local
topological invariants. Consequently, proposals for the experimental detection
using local probes are rare. A recent paper [Slager et al., Phys. Rev. B 92,
085126 (2015)] has argued, based on theoretical results for a particular class
of models, that insulators with topologically trivial and non-trivial band
structures in two space dimensions display a qualitatively different response
to point-like impurities. Here we present a comprehensive investigation of the
impurity response of a large set of models of non-interacting electrons on the
honeycomb lattice, driven insulating by either broken inversion, broken time
reversal, broken $C_3$, or broken translation symmetry. These cases include
Hofstadter bands, strain-induced pseudo-Landau levels and higher-order
topological insulators. Our results confirm that for hopping models respecting
the lattice symmetries, the response to a single impurity can indeed
distinguish between trivial and non-trivial band topology. However, for
modulated or inhomogeneous host systems we find that trivial states of matter
can display an impurity response akin to that of topologically non-trivial
states, and thus the diagnostic fails.
