Most often than not, the sheaves appearing in algebraic geometry (with the Zariski topology) are $\mathcal{O}_X$-modules, instead of simple abelian sheaves.

Now, when dealing with topological spaces (for example, in Verdier duality) and in étale cohomology, it seems that abelian sheaves have the main role.

I wonder why. For example, Verdier duality works just fine as a statement about $\mathcal{O}_X$-modules (as in Spaltenstein's *Resolution of unbounded complexes*) and we recover the standard topological statements setting $\mathcal{O}_X=\underline{\mathbb{Z}}$. Even more, since ringed spaces always have fibered products (and the underlying topological spaces are the usual fibered products), base change also works well.

(I'm not sure about the étale case. I would love to know more about it.)

Is there some fundamental reason *not* to consider $\mathcal{O}_X$-modules in these cases?