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We describe a natural decomposition of a normal complex surface singularity$(X,0)$ into its "thick" and "thin" parts. The former is essentially metricallyconical, while the latter shrinks rapidly in thickness as it approaches theorigin. The thin part is empty if and only if the singularity is metricallyconical; the link of the singularity is then Seifert fibered. In general thethin part will not be empty, in which case it always carries essentialtopology. Our decomposition has some analogy with the Margulis thick-thindecomposition for a negatively curved manifold. However, the geometric behavioris very different; for example, often most of the topology of a normal surfacesingularity is concentrated in the thin parts.  By refining the thick-thin decomposition, we then give a complete descriptionof the intrinsic bilipschitz geometry of $(X,0)$ in terms of its topology and afinite list of numerical bilipschitz invariants.

The thick-thin decomposition and the bilipschitz classification of normal surface singularities