Non-quasi-coherent sheaves of $\mathcal O\_X$ modules on a scheme seem like a wild concept to me; are they actually used for something? 



When are non-quasi-coherent sheaves used?
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 Riemann-Roch. I'm not sure when this was reinterpreted in terms of sheaves, which were only introduced later.


