One can construct $(S,s)$ and $\mathscr{Y}$ satisfying (1) and (2) by means of a standard argument based on spreading out and Greenberg approximation; see for instance [MN15, 5.1.2] and the proof of [NX16a, 4.2.4] (note that normality of $\mathscr{Y}$ automatically follows from the fact that $\mathscr{Y}\setminus\mathscr{Y}_{s}$ is normal and $\mathscr{Y}_{s}\cong\mathscr{X}_{k}$ is reduced). In this construction we can also spread out a global generator $\omega$ of the relative canonical line bundle $\omega_{\mathscr{X}/R}$, which then induces a global generator for $\omega_{\mathscr{Y}/S}$, yielding the triviality of $K_{\mathscr{Y}/S}$.

If $N$ is at least $2$, then for every point $x$ of $\mathscr{X}_{k}\cong\mathscr{Y}_{s}$, the model $\mathscr{X}$ is regular at $x$ if and only if $\mathscr{Y}$ is regular at $x$ [MN15, 5.1.2(d)]. Thus the pair $(\mathscr{Y},\mathscr{Y}_{s})$ is snc at all the points of $\mathscr{Y}_{s}$ where $(\mathscr{X},\mathscr{X}_{k})$ is snc. Taking $N$ sufficiently large, we can arrange that every prime component $E$ of $\mathscr{Y}_{s}$ is $\mathbb{Q}$-Cartier in $\mathscr{Y}$. More precisely, let $x$ be a point of $\mathscr{X}_{k}$ and let $m_{x}$ be the Cartier index of $E$ in $\mathscr{X}$ at $x$. Let $f$ be a local generator for the ideal sheaf $\mathcal{O}_{\mathscr{X}}(-m_{x}E)$ at $x$. Assume that $N>m_{x}$ and let $g$ be any element of $\mathcal{O}_{\mathscr{Y},x}$ that is congruent to $f$ modulo $t^{N}$. Obviously, $g$ cannot vanish at any other component of $\mathscr{Y}_{s}$, because $t$ vanishes along each of these components and $f$ does not. On the other hand, $f$ divides $t^{m_{x}}$ in $\mathcal{O}_{\mathscr{X},x}$, so that $g$ divides $t^{m_{x}}$ in $\mathcal{O}_{\mathscr{Y},y}$ since $N>m_{x}$. Thus the zero locus of $g$ is supported in $\mathscr{Y}_{s}$, which means that $g=0$ is a local equation for $m_{x}E$ in $\mathscr{Y}$ at $x$. From now on, we assume that $N$ has been chosen large enough to guarantee that $N\geq 2$ and every prime component of $\mathscr{Y}_{s}$ is $\mathbb{Q}$-Cartier.

Let $E$ be a prime component of $\mathscr{X}_{k}$, denote by $\widetilde{E}$ its normalization, and let $\Delta$ be the pullback of the $\mathbb{Q}$-Cartier divisor $\mathscr{X}_{k}-E$ to $\widetilde{E}$. The scheme $\mathscr{X}$ is regular in codimension two by Lemma 4.2. It follows that the different $\mathrm{Diff}_{\widetilde{E}}(\mathscr{X}_{k}-E)$ coincides with $\Delta$. Thus the pair $(\widetilde{E},\Delta)$ is dlt by adjunction [Ko13, 4.8], using the same reasoning as in the proof of [Ko13, 4.16.4] (except that we have not yet established the normality of $E$). Since $N\geq 2$, the scheme $\mathscr{Y}$ is regular in codimension two, as well; since it is of finite type over $k$, we can apply inversion of adjunction [Ko13, 4.9] to deduce that $(\mathscr{Y},\mathscr{Y}_{s})$ is log canonical on a neighbourhood of $E$, and that the log canonical centers of $(\mathscr{Y},\mathscr{Y}_{s})$ contained in $E$ are precisely the images of the log canonical centers of $(\widetilde{E},\Delta)$. At the generic point of such a log canonical center, the pair $(\mathscr{Y},\mathscr{Y}_{s})$ is snc because the same holds for $(\mathscr{X},\mathscr{X}_{k})$. Varying $E$, we obtain that $(\mathscr{Y},\mathscr{Y}_{s})$ is dlt. This implies that every stratum of $\mathscr{X}_{k}\cong\mathscr{Y}_{s}$ is normal [Ko13, 4.16]; thus, in retrospect, we see that $\widetilde{E}=E$. ∎