  In this paper we prove a result on the effective generation of
pluri-canonical linear systems on foliated surfaces of general type. Fix a
function $P: \mathbb Z_{\geq 0}\to \mathbb Z $, then there exists an integer
$N_1>0$ such that if $(X,\mathcal F)$ is a canonical or nef model of a
foliation of general type with Hilbert polynomial $\chi (X, mK_{\mathcal
F})=P(m)$ for all $m\in \mathbb Z_{\geq 0}$, then $|mK_{\mathcal F}|$ defines a
birational map for all $m\geq N_1$.
  We also prove a Grauert-Riemannschneider type vanishing theorem for foliated
surfaces with canonical singularities.
