  We prove that for every smooth prime Fano $3$-fold $V$, the Hilbert scheme
$\operatorname{Hilb}^{sc} V$ of smooth connected curves on $V$ contains a
generically non-reduced irreducible component of Mumford type. We also study
the deformations of degenerate curves $C$ in $V$, i.e., curves $C$ contained in
a smooth anti-canonical member $S \in |-K_V|$ of $V$. We give a sufficient
condition for $C$ to be stably degenerate, i.e., every small (and global)
deformation of $C$ in $V$ is contained in a deformation of $S$ in $V$. As a
result, by using the Hilbert-flag scheme of $V$, we determine the dimension and
the smoothness of $\operatorname{Hilb}^{sc} V$ at the point $[C]$, assuming
that the class of $C$ in $\operatorname{Pic} S$ is generated by
$-K_V\big{\vert}_S$ together with the class of a line, or a conic on $V$.

Algebraic Geometry