Let $Xsubset mathbb{P}^4$ 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 $Xsubset mathbb{P}^4$. A singular point on such a hypersurface is either of type $cA_n$ ($ngeq 1$), or of type $cD_m$ ($mgeq 4$), 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 $2leq nleq 7$ (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 $8$.
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