Take a smooth manifold $M^n$ with a smooth foliation $F$. Consider the sheaf $\cal F$ of $C^{\infty}$ functions on $M^n$, locally constant along the foliation $F$. What is known about Chech cohomology of such a sheaf?

I am pretty sure that such a question was studied (and maybe even has a complete answer), but I don't know a reference.

A more specific question is: what happen when $F$ is 1-dimensional, given by integral trajectories of a non-vanishing vector field? Or even more specifically, suppose $H^1(M^n)=0$ and we consider a Killing vector field $v$ on $M^n$ (i.e. $v$ is preserving a metric). Is it true the the sheaf of functions $\cal F$ locally constant along trajectories of $v$ is acyclic? (we need $H^1(M^n)=0$, otherwise $S^1$ will be an obvious counterexample).