Let $X$ be a ringed space. A quasi-coherent module on $X$ is a module which has locally a presentation, i.e. locally on $X$, it is the cokernel of a map between free modules. If $X$ is a scheme, then the category of quasi-coherent modules on $X$ or its derived category are quite important to understand $X$. In the stacks project (modules) you can find some theorems about quasi-coherent modules on arbitrary ringed spaces. Now I wonder if this is just abstract nonsense:
Question: Are there nontrivial, interesting examples of quasi-coherent modules on ringed spaces, which are no schemes? Or do they play no role outside algebraic geometry?
Of course, locally free modules of finite rank correspond to vector bundles and are important in other geometries as well (for example, manifolds when regarded as locally ringed spaces). Are there other quasi-coherent modules of interest?