In a course on measure theory, the lecturer proved that distributions (on a locally convex space I think) form a sheaf $\mathcal D$. He isn't interested in sheaves, so he didn't elaborate. Afterwards, I looked online for material on this sheaf, but didn't really find any information about it. On MSE, there is this question regarding sheaf cohomology w.r.t to $\mathcal D$, which looks cool but is beyond me.
What are some interesting properties of $\mathcal D$? Is it soft, flabby, or fine? What else?