Let
$X\subset \mathbb{P}^{4}$
be a terminal factorial quartic
$3$
-fold. If
$X$
is non-singular,
$X$
is birationally rigid, i.e. the classical minimal model program 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 of type
$cA_{n}$
(
$n\geqslant 1$
), or of type
$cD_{m}$
(
$m\geqslant 4$
) 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
$2\leqslant n\leqslant 7$
, (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
$8$
.