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.



Is there a known classification of regular multiplicity-free permutation groups?
These are the abelian regular permutation groups. The permutation character in this case is the character of the regular representation and in the regular representation a character appears with multiplicity equal to the dimension of the irreducible representation. So it can be multiplicity free iff all irreducibles are one dimensional which is equivalent to abelian.


