  We investigate the inhomogeneous Rashba chain coupled to a superconducting
substrate, hosting the Majorana quasiparticles near its edges. We discuss its
subgap spectrum and study how robust are the zero-energy quasiparticles against
the diagonal and off-diagonal disorder. Studying the $\mathbb{Z}_2$ topological
invariant we show that disorder induced transition from the topologically
non-trivial to trivial phases is manifested by characteristic features in the
spatially-resolved quasiparticle spectrum at zero energy. We provide evidence
for the non-local nature of the zero-energy Majorana quasiparticles, that are
well preserved upon partitioning the chain into separate pieces. Even though
the Majorana quasiparticles are not completely immune to inhomogeneity we show
that they can spread onto other (normal) nanoscopic objects via the proximity
effect.
