  This paper delves into the study of curvilinear Hilbert schemes associated
with a singular variety $(X,0)$ and the relationship between their motivic
classes and the motivic measure on the arc scheme $X_\infty$ of $X$ introduced
by Denef and Loeser. We introduce an Igusa zeta function specifically tailored
for curvilinear Hilbert schemes for which we provide an explicit formulation in
terms of an embedded resolution of the singularity, and we consequently obtain
a recursive formula to compute the motivic classes of curvilinear Hilbert
schemes in terms of the resolution. In addition, the paper explores and
analyzes the geometry and combinatorics of curvilinear Hilbert schemes in the
context of plane curve singularities and their topological invariants.
