  Let $X\subset \mathbb{P}^4$ be a terminal factorial quartic $3$-fold. If $X$
is non-singular, $X$ is \emph{birationally rigid}, i.e. the classical MMP on
any terminal $\mathbb{Q}$-factorial projective variety $Z$ birational to $X$
always terminates with $X$. This no longer holds when $X$ is singular, but very
few examples of non-rigid factorial quartics are known. In this article, we
first bound the local analytic type of singularities that may occur on a
terminal factorial quartic hypersurface $X\subset \mathbb{P}^4$. A singular
point on such a hypersurface is either of type $cA_n$ ($n\geq 1$), or of type
$cD_m$ ($m\geq 4$), or of type $cE_6, cE_7$ or $cE_8$. We first show that if
$(P \in X)$ is of type $cA_n$, $n$ is at most $7$, and if $(P \in X)$ is of
type $cD_m$, $m$ is at most $8$. We then construct examples of non-rigid
factorial quartic hypersurfaces whose singular loci consist (a) of a single
point of type $cA_n$ for $2\leq n\leq 7$ (b) of a single point of type $cD_m$
for $m= 4$ or $5$ and (c) of a single point of type $cE_k$ for $k=6,7$ or $8$.
