In this talk I will explain a work in progress, joint with Baldur Sigurdsson, in which we explore an idea by A'Campo. An explicit collapsing map is constructed for each isolated hypersurface singularity using a natural connection depending on the ambient metric. The preimage of the singular point by the collapsing map yields, on each Milnor fiber, a piecewise linear spine. The combinatorics and the properties of this spine are analyzed by means of a vector field which is defined on the boundary of the real oriented blow-up of a resolution of the singularity that also resolves the polar curves. At the moment, we deal with the case of isolated plane curve singularities.