We first use absolute Noetherian approximation to reduce to the case of schemes of finite type over $\mathbf{Z}$. The question is local on $X$ and $S$. Hence we may assume that $X$ and $S$ are affine. Then we can write $S = \Spec(R)$ and write $R$ as a filtered colimit $R = \colim R_i$ of finite type $\mathbf{Z}$-algebras. Using Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find an $i$ and a morphism $f_i : X_i \to \Spec(R_i)$ whose base change to $S$ is $f$. After increasing $i$ we may assume that $f_i$ is at-worst-nodal of relative dimension $1$, see Lemma \ref{lemma-descend-nodal-family}. The image $x_i \in X_i$ of $x$ will be a singular point of its fibre, for example because the formation of $\text{Sing}(f)$ commutes with base change (Divisors, Lemma \ref{divisors-lemma-base-change-and-fitting-ideal-omega}). If we can prove the lemma for $f_i : X_i \to S_i$ and $x_i$, then the lemma follows for $f : X \to S$ by base change. Thus we reduce to the case studied in the next paragraph. \medskip\noindent Assume $S$ is of finite type over $\mathbf{Z}$. Let $s \in S$ be the image of $x$. Recall that $\kappa(x)$ is a finite separable extension of $\kappa(s)$, for example because $\text{Sing}(f) \to S$ is unramified or because $x$ is a node of the fibre $X_s$ and we can apply Lemma \ref{lemma-nodal}. Furthermore, let $\kappa'/\kappa(x)$ be the degree $2$ separable algebra associated to $\mathcal{O}_{X_s, x}$ in Remark \ref{remark-quadratic-extension}. By More on Morphisms, Lemma \ref{more-morphisms-lemma-realize-prescribed-residue-field-extension-etale} we can choose an \'etale neighbourhood $(V, v) \to (S, s)$ such that the extension $\kappa(v)/\kappa(s)$ realizes either the extension $\kappa(x)/\kappa(s)$ in case $\kappa' \cong \kappa(x) \times \kappa(x)$ or the extension $\kappa'/\kappa(s)$ if $\kappa'$ is a field. After replacing $X$ by $X \times_S V$ and $S$ by $V$ we reduce to the situation described in the next paragraph. \medskip\noindent Assume $S$ is of finite type over $\mathbf{Z}$ and $x \in X_s$ is a split node, see Definition \ref{definition-split-node}. By Lemma \ref{lemma-formal-local-structure-nodal-family} we see that there exists an $\mathcal{O}_{S, s}$-algebra isomorphism $$ \mathcal{O}_{X, x}^\wedge \cong \mathcal{O}_{S, s}^\wedge[[s, t]]/(st - h) $$ for some $h \in \mathfrak m_s^\wedge \subset \mathcal{O}_{S, s}^\wedge$. In other words, if we consider the homomorphism $$ \sigma : \mathbf{Z}[a] \longrightarrow \mathcal{O}_{S, s}^\wedge $$ sending $a$ to $h$, then there exists an $\mathcal{O}_{S, s}$-algebra isomorphism $$ \mathcal{O}_{X, x}^\wedge \longrightarrow \mathcal{O}_{Y_\sigma, y_\sigma}^\wedge $$ where $$ Y_\sigma = \Spec(\mathbf{Z}[u, v, t]/(uv - a)) \times_{\Spec(\mathbf{Z}[a]), \sigma} \Spec(\mathcal{O}_{S, s}^\wedge) $$ and $y_\sigma$ is the point of $Y_\sigma$ lying over the closed point of $\Spec(\mathcal{O}_{S, s}^\wedge)$ and having coordinates $u, v$ equal to zero. Since $\mathcal{O}_{S, s}$ is a G-ring by More on Algebra, Proposition \ref{more-algebra-proposition-ubiquity-G-ring} we may apply More on Morphisms, Lemma \ref{more-morphisms-lemma-relative-map-approximation-pre} to conclude.