The local topology of isolated complex surface singularites is long understood, as cones on closed 3-manifolds obtained by negative definite plumbing. On the other hand a full understanding of the analytic types is out of reach, motivating Zariski's efforts into the 1980's to give a good concept of "equisingularity" for families of singularities.
The significance of Lipschitz geometry as a tool in singularity theory is a recent insight, starting (in complex dimension 2) with examples of Birbrair and Fernandes published in 2008. I will describe work with Anne Pichon proving that Zariski equisingularity in complex dimension two (and lower) is equivalent to constant Lipschitz geometry. This builds on earlier joint work with Birbrair and Pichon on classifying the inner geometry in terms of discrete data associated with a refined JSJ decomposition of the associated 3-manifold link.