  Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class
sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing
element} if there exists an irreducible character of $G$ taking the value $0$
on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$
whose elements are the conjugacy class sizes of the vanishing elements of $G$.
Motivated by the results in \cite{BLP}, we describe the class of the finite
groups $G$ such that vcs$(G)$ consists of a single element \emph{under the
assumption that $G$ is supersolvable or $G$ has a normal Sylow $2$-subgroup}
(in particular, groups of odd order are covered). As a particular case, we also
get a characterization of finite groups having a single vanishing conjugacy
class size \emph{which is either a prime power or square-free}.
