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