When doing concrete calculations of the sheaf cohomology groups $H^k(X;\mathcal F)$ for some topological space $X$ and some sheaf $\mathcal F$ on $X$, it is usually easiest to make use of Leray coverings. However, sometimes one can prove a statement of the form "for any open covering $U$ of $X$, $H^k(U;\mathcal F)$ equals [something]". Then, if one knows there exists a Leray covering, this suffices to determine $H^k(X;\mathcal F)$. This leads me to ask the following question:
Are there any concrete criteria to determine whether a sheaf $\mathcal F$ on $X$ admits a Leray covering?
In particular, is there always a Leray covering? I am particularly interested in the case where $X$ is a complex manifold, or even a Riemann surface; I don't know if this makes things easier.