Question: Which projective algebraic surfaces admit algebraic foliations?
I realize that the question is a bit too general. There is a classification of minimal surfaces – not all the surfaces. And I do not think, one can say something about foliations after blow up. But I am looking for at least partial results.
More concrete question may be: is there a surface, admitting a foliation, in each birational class.
Let me state Brunella's theorem that describes the regular foliated surfaces.
Thm: Let $X$ ba a complex projective surface and let $\mathcal{F}$ be a holomorphic foliation without singularities on $X$. Then we fall in one of the following cases:
1) $\mathcal{F}$ is a fibration;
2) $\mathcal{F}$ is trabnsverse to a fibration, hence it is the suspension of an automorphism group of an algebraic curve;
3) $\mathcal{F}$ in a turbulent foliation on an elliptic surface;
4) $\mathcal{F}$ is a linear foliation on a complex torus;
5) $\mathcal{F}$ a transversely hyperbolic foliation with dense leaves whose universal cover is a disk fibration over the disk.
A simple obstruction to the existence of regular foliations with tangent bundle $L$ on a surface $X$ is the vanishing of the Chern class $$ c_2(TX-L)$$
This measures the number of singularities by Baum-Bott's Theorem.
Another way to see this is that $X$ admits a regular foliation if and only if $TX$ has a holomorphic sub-line-bundle.