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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.

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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 free 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 $Y$.

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Suppose that $f:X\to Y$ 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 , which is semi-stableand let $D\subseteq X$ be a normal crossings divisor. Suppose also that $Y$ f$is semi-stable relatively to$E,D$and$k$, in the spectrum sense of a Dedekind domainIllusie in par. 1.4 of "Réduction semi-stable et décomposition...", Duke Math. J. 60 (1990). The morphism$f$is then flat and lci and its fibres are reduced normal crossings divisors. There is then a relative residue sequence $$0\to \Omega_{X/Y}\to\Omega_{X/Y}({\rm log})\to{\cal O}_{D_0}\to log})\to F\to 0\ \ \ \ (*)$$ where$D_0$F$ is the direct sum of the structure sheaves of supported on the irreducible components 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 $$\dots\to {\rm Tor}^1_Y(\Omega_{X/Y},M)\to{\rm Tor}^1_Y(\Omega_{X/Y}({\rm log}),M)\to{\rm Tor}^1_Y({\cal O}_{D_0},MTor}^1_Y(F,M)$$ $$\to \Omega_{X/Y}\otimes_Y M\to\Omega_{X/Y}({\rm log})\otimes_Y M\to{\cal O}_{D_0}\otimes_Y M\to F\otimes_Y M\to 0$$ 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 free 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 $Y$.

I am writing this in a rush, so my apologies for any silly mistakes.

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