  Let $n\ge 2$. In this paper we study some properties of the virtual singular
braid group $VSG_n$. We define some numerical invariants for virtual singular
braids arising from exponents of words in $VSG_n$ and describe the kernel of
these homomorphisms. We determine all possible homomorphisms, up to
conjugation, from the virtual singular braid group $VSG_n$ to the symmetric
group $S_n$, and for the particular case $n=2$ we give a presentation (and a
description) for the kernel in each case. For all possible homomorphisms we
obtain a decomposition of $VSG_n$ as a semi-direct product of the kernel of the
homomorphism and the symmetric group. Finally, we show that there are some
forbidden relations in $VSG_n$ and then we introduce some quotients of it (e.g.
the welded singular braid group, the unrestricted virtual singular braid group,
among other related groups) and then we state for them similar results obtained
for $VSG_n$.
