I am going to describe a decomposition of a germ of a complex algebraic surface near an isolated singular point, such that this decomposition gives a complete invariant of this germ with respect to local bi-Lipschitz homeomorphisms.  This decomposition  is related to JSJ Decomposition of the link of this singularity, considered as a smooth 3-manifold.  The construction uses the carousel decomposition of neighborhoods of complex algebraic curves, constructed by Le Dung Trang.  It is a joint work with Walter Neumann and Anne Pichon.