Suppose that $Y$ is the spectrum of a smooth curve over perfect field $k$ and let
$D\subseteq Y$ be a finite set of closed points. Let $f:X\to Y$
be a proper morphism and let $D\subseteq X$ be a normal crossings divisor. Suppose
that $f$ is semi-stable relatively to $E,D$ and $k$, in the sense of
Illusie in par. 1.4 of "Réduction semi-stable et décomposition...", Duke Math. J. 60 (1990).

The morphism $f$ is then flat and lci and its fibres are reduced normal crossings divisors. There is a relative residue sequence
$$
0\to \Omega_{X/Y}\to\Omega_{X/Y}({\rm log})\to F\to 0\ \ \ \ (*)
$$
where $F$ is supported on the singular locus of the singular fibres of $f$, and $\Omega_{X/Y}({\rm log})$ is the locally free sheaf of differentials with (relative) logarithmic singularities along $D$.
See for instance p. 23 in "Une conjecture sur la torsion..." by V. Maillot and D. Rössler
(Publ. Res. Inst. Math. Sci. 46, no. 4 (2011) - for lack of a canonical reference (?)).

Now let $M$ be any quasi-coherent ${\cal O}_Y$-module.
The tor-sequence corresponding to $\otimes_Y M$ when applied to (*) gives
$$
\dots\to {\rm Tor}^1_Y(\Omega_{X/Y},M)\to{\rm Tor}^1_Y(\Omega_{X/Y}({\rm log}),M)\to{\rm Tor}^1_Y(F,M)
$$
$$
\to \Omega_{X/Y}\otimes_Y M\to\Omega_{X/Y}({\rm log})\otimes_Y M\to F\otimes_Y M\to 0
$$
and since ${\rm Tor}^l_Y(\Omega_{X/Y}({\rm log}),M)=0$ for all $l>0$ (because $\Omega_{X/Y}({\rm log})$ is locally free and $f$ is flat) and
${\rm Tor}^l_Y(N,K)=0$ for any $l>1$ and any quasi-coherent ${\cal O}_Y$-modules $N,K$
(that is because $Y$ is the spectrum of a Dedekind domain and any finitely generated quasi-coherent
${\cal O}_Y$-module has a two-step projective resolution; the general case follows from compatibility of Tor with direct limits), we see that
${\rm Tor}^l_Y(\Omega_{X/Y},M)=0$, for all $l>0$, ie $\Omega_{X/Y}$ is flat over $Y$.

EDIT As remarked by Liu below, the sheaf $\Omega_{X/Y}$ can also be seen to be flat simply because it is the subsheaf of a torsion free sheaf.