  We study mesoscopic signatures of the topological Anderson transitions in
topological disordered chains. To this end we introduce an integer-valued
sample-specific definition of the topological index in finite size systems. Its
phase diagram exhibits a fascinating structure of intermittent topological
phases, dubbed topological islands. Their existence is rooted in the real zeros
of the underlying random polynomial. Their statistics exhibits finite-size
scaling, pointing to the location of the bulk topological Anderson transition.
While the average theories in AIII and BDI symmetry classes are rather similar,
the corresponding patterns of topological islands and their statistics are
qualitatively different. We also discuss observable signatures of sharp
topological transitions in mesoscopic systems, such as persistent currents and
entanglement spectra.
