  We show that the Zariski canonical stratification of complex hypersurfaces is
locally bi-Lipschitz trivial along the strata of codimension two. More
precisely, we study Zariski equisingular families of surface, not necessarily
isolated, singularities in $\mathbb{C}^3$. We show that a natural
stratification of such a family given by the singular set and the generic
family of polar curves provides a Lipschitz stratification in the sense of
Mostowski. In particular, such families are bi-Lipschitz trivial by
trivializations obtained by integrating Lipschitz vector fields.
