Let $(R,m)$, $(S,n)$ be two local Noetherian domains, $R \subseteq S$ is flat, and $m \subseteq n$.
Question 1: If $R$ is regular and $S$ is Cohen-Macaulay, is $S$ also regular?
Question 2: If $R$ is Cohen-Macaulay and $S$ is regular, is $R$ also regular?
My favorite examples: $R=k[x(x-1)]_{(x(x-1))}$, $S=k[x]_{(x)}$, $R \subseteq S$ is flat (as a localization of a free extension) and both local domains are regular (localization of a polynomial ring).
$R=k[x^2,x^3]_{(x^2,x^3)}$, $S=k[x]_{(x)}$, but the extension is not flat.
Notice that $R=k[x^2]_{(x^2)}$, $S=k[x^2,x^3]_{(x^2,x^3)}$ is flat, $R$ is regular, $S$ is CM, but not regular.
I guess that in some special cases one can apply the miracle flatness theorem. I do not want to assume that $mS$ is a prime ideal in $S$. I do not mind to assume that $\dim(R)=\dim(S)$.
Thank you very much!
Edit: Perhaps the second answer to this question may help. It says: "If $f: A \to B$ is flat, then obviously the image of any $A$-regular sequence under $f$ is a $B$-regular sequence." (Does it answer Question 1?).