  Recently, it is shown that each regular homogeneous Finsler space $M$ admits
at least one homogeneous geodesic through any point $o\in M$. The purpose of
this article is to study the existence of homogeneous geodesics on singular
homogeneous $(\alpha,\beta)$-spaces, specially, homogeneous Kropina spaces. We
show that any homogeneous Kropina space admits at least one homogeneous
geodesic through any point. It is shown that, under some conditions, the same
result is true for any $(\alpha,\beta)$-homogeneous space. Also, in the case of
homogeneous Kropina space of Douglas type, a necessary and sufficient condition
for a vector to be a geodesic vector is given. Finally, as an example,
homogeneous geodesics of $3$-dimensional non-unimodular real Lie groups
equipped with a left invariant Randers metric of Douglas type are investigated.
