Cohomology of a sheaf of functions locally constant along a foliation

I am joining this discussion a bit late, but let me add an example. If you consider a smooth minimal action of Z on the circle S^1 the suspension gives a flow on the torus. If the action is C^2 conjugate to an irrational rotation, then the transverse basic cohomology is finite dimensional. But if the action is only topologically conjugate to a rotation, then the basic cohomology may be infinite. The literature on this is quite a long time ago, in the 1970's perhaps. here is one reference

Haefliger, A.and Banghe, Li
Currents on a circle invariant by a Fuchsian group. Geometric dynamics (Rio de Janeiro, 1981), 369–378, Lecture Notes in Math., 1007, Springer, Berlin, 1983.

Here is a more recent article

Avila, Artur and Kocsard, Alejandro
Cohomological equations and invariant distributions for minimal circle diffeomorphisms. Duke Math. J. 158 (2011), no. 3, 501–536.

and there is one more artcile that is likely relevant to the question

Lott, John
Invariant currents on limit sets. Comment. Math. Helv. 75 (2000), no. 2, 319–350.