  Smale-Barden manifolds $M$ are classified by their second homology
$H_2(M,{\mathbb Z})$ and the Barden invariant $i(M)$. It is an important and
dificult question to decide when $M$ admits a Sasakian structure in terms of
these data. In this work we show methods of doing this. In particular we
realize all $M$ with $H_2(M)={\mathbb Z}^k\oplus(\oplus_{i=1}^r{\mathbb
Z}_{m_i}^{2g_i})$ and $i=0,\infty$, provided that $k\geq 1$, $m_i\geq 2$,
$g_i\geq 1$, $m_i$ are pairwise coprime. Using our methods we also contribute
to the problem of the existence of definite Sasakian structures on rational
homology spheres. Also, we give a complete solution to the problem of the
existence of Sasakian structures on rational homology spheres in the class of
semi-regular Sasakian structures.
