The topological stability introduced by \textit{P. Walters} [Lect. Notes Math. 668, 231-244 (1978; Zbl 0403.58019)] is a kind of structural stability defined for all homeomorphisms of compact metric spaces. Under this assumption, \textit{P. Walters} proved that topologically stable homeomorphisms satisfy the shadowing property. The main part of the paper is concerned with the relation between topological stability and shadowing properties for zero-dimensional dynamical systems. A conjecture by \textit{E. Akin} et al. [Trans. Am. Math. Soc. 360, No. 7, 3613--3630 (2008; Zbl 1144.22007)] states that if a homeomorphism of a closed topological manifold is topologically stable, then its restriction to the non-wandering set is expansive. This conjecture fails when the space is totally disconnected. Here the author shows that if the dimension of the space is zero, topologically stable homeomorphisms may exhibit a non-hyperbolic behavior. The author also observes some singular behaviors when the spaces are manifolds. He proves that those homeomorphisms satisfy the strict periodic shadowing property by a modification of Walters' argument. Several examples and counterexamples are presented. Furthermore, the author proves that any topologically stable (in a modified sense) homeomorphism of a Cantor space exhibits only simple typical dynamics.