For any locally ringed space $\left(X,O_X\right)$, a quasicoherent sheaf is a sheaf of $O_X$-modules which are locally the quotient of free modules.
Considering a manifold (Haussdorf, second-countable) as a locally ringed space, what are the quasi-coherent sheaves? Is there a good characterisation (some Serre-Swan type generalisation)? What are some examples which are not vector bundles?
The answers to this question only talks about coherent sheaves and why they are not interesting. I am interested in the answer for quasicoherent sheaves.