Assume that $S$ is Artinian local and $f$ is proper and flat. If $H^0(X, \mathcal{O}_X)\otimes k(s)\to H^0(X_s, \mathcal{O}_{X_s})$ is surjective then Theorem 5.10 (FGA Explained) implies that $H^0(X, \mathcal{O}_X)$ is a flat module over $H^0(S, \mathcal{O}_S)$. However that is not always true https://mathoverflow.net/a/107603/158636

Assume that $S$ is Artinian local and $f$ is proper and flat. If $H^0(X, \mathcal{O}_X)\otimes k(s)\to H^0(X_s, \mathcal{O}_{X_s})$ is surjective then standard cohomology and base change results imply that $H^0(X, \mathcal{O}_X)$ is a flat module over $H^0(S, \mathcal{O}_S)$. However that is not always true https://mathoverflow.net/a/107603/158636