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.
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