Abstract: Let $(X,0)\subset(\mathbbm{C}^N,0)$ be a germ of complex singularity. The hermitian metric of $\mathbbm{C}^N$ induces two metrics on $(X,0)$ called the outer and the inner metric. The bilipschitz type of these metrics is independent of the choice of embedding. It is well known that the topology of $(X,0)$ is a cone over the link. A natural question is then when is the inner (or outer) metric bilipschitz equivalent to a metric cone? We will give a complete answer to this question when $(X,0)$ is a rational surface singularity, using the thick-thin decomposition of Birbrair, Neumann and Pichon. We will then restrict ourselves to minimal singularities, and describe the complete classification of their inner (and outer) metric, and especially show that it is determined by the topology.