Let $X$ be a paracompactHausdorff space. A sheaf$F$ of groups over $X$ is fine if for every two disjoint closed subsets $A,B\subset X$, $A\cap B = \emptyset$, there is an endomorphism of the sheaf of groups $F\to F$ which restricts to the identity in a neighborhood of $A$ and to the $0$ endomorphism in a neighborhood of $B$. Every fine sheaf is soft.

A slightly different definition is given in Voisin 2002 (Vol I, Def. 4.35):

A fine sheaf$\mathcal{F}$ over $X$ is a sheaf of $\mathcal{A}$-modules, where $\mathcal{A}$ is a sheaf of rings such that, for every open cover$U_i$ of $X$, there is a partition of unity$1 = \sum f_i$ (where the sum is locally finite) subordinate to this covering.

Another definition is given by Godement 1958 (the paragraph after Theorem II.3.7.2): a sheaf $F$ is fine if the internal hom$Hom(F,F)$ is a soft sheaf.

Properties

For any fine sheaf $J$ over a paracompact Hausdorff space $M$, the sheaf cohomology groups of positive degree $j$ are all trivial