We will explore certain $C^0$-rigidity and flexibility phenomena in the study of contact transformations. In particular, we will show how the dichotomy between contact squeezing and non-squeezing is related to the Rokhlin property of the group of contact homeomorphisms. Assuming a more quantitative viewpoint, we will define new distances on the group of contact homeomorphisms and show that, in some cases, they satisfy a form of $C^0$-stability. A technical result behind this stability is $C^0$-continuity of Sandon's spectral invariants. The talk is based on a joint work with Baptiste Serraille.