  Pfister and Steenbrink studied punctual Hilbert schemes for irreducible curve
singularities. In particular, they investigated the structure of special
punctual Hilbert schemes for certain monomial curve singularities. In this
paper, we study the punctual Hilbert schemes of all degrees for the curve
singularities of types $E_6$ and $E_8$. For our analysis, we introduce
computational algorithms to decompose a punctual Hilbert schemes into affine
cells. We also use the theory of Gr\"obner basis and known results about the
compactified Jacobian of singular curves to prove our main theorems.
