  We begin with a comprehensive discussion of the punctual Hilbert scheme of
the regular two-dimensional local ring in terms of the Gr\"obner cells. These
schemes are the most degenerate fibers of the Grothendieck-Deligne norm map
(the Hilbert-Chow morphism), playing an important role in the study of Hilbert
schemes of smooth surfaces. They are generally singular, but their Gr\"obner
cells are affine spaces; they admit an explicit parametrization due to Conca
and Valla. We use this to obtain the Gr\"obner decomposition of compactified
Jacobians of plane curve singularities, which is non-trivial even for the
generalized Jacobians (principal ideals only). One of the application is the
topological invariance of certain variants of compactified Jacobians and the
corresponding motivic superpolynomials for analytic deformations of
quasi-homogenous plane curve singularities and some similar families.
