The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$.
- $\Sigma$ is regular if it acts transitively and freely on $\Omega$, i.e., for any two $i,j\in \Omega$ there is a unique $\sigma\in\Sigma$ with $\sigma(i)=j$.
- $\Sigma$ is multiplicity-free if its permutation character (the character of the linear representation of $\Sigma$ by permutation matrices) is the sum of distinct irreducible characters.
Examples are the permutation groups generated by a single cyclic permutation.