Nonquasicoherent sheaves of $\mathcal O_X$ modules on a scheme seem like a wild concept to me; are they actually used for something?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

One can think the adeles on a curve (or higher adeles on other spaces) as a sheaf of $\mathcal O$algebras. That is, consider the sheaf $B(U)=\prod_{x\in U}\mathcal O_x$, where $\mathcal O_x$ is the completion of $\mathcal O$. Then the sheaf $A=B\otimes K$, where $K$ is the sheaf of rational functions, has the adeles as global sections. There is a short exact sequence $\mathcal O\to K\times B\to A$. One can tensor a quasicoherent sheaf with this to obtain a resolution to compute cohomology. Indeed, Weil introduced the adeles (after the earlier ideles) specifically to prove RiemannRoch. I'm not sure when this was reinterpreted in terms of sheaves, which were only introduced later. 

