Suppose, $(R,m)$ is a Noetherian ring that is not a domain. Can $\hat{R}_m$ be a domain?
I think this cannot be the case for if $a,b\in R$ s.t. $a,b\neq 0$ and $ab=0$. Then, this $R$ is a Noetherian local ring, by the Krull intersection theorem, $\cap_i m^i=0$ and the LHS of this is the kernel of the canonical map $R\to \hat{R}_m$. So this map is injective. So $(a,a,a,...)$ and $(b,b,b,...)$ are non-zero elements of $\hat{R}_m$ whose product of zero.