TITLE: Examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves
QUESTION [19 upvotes]: Let $X = \operatorname{Spec} k[x]_{(x)}$ which consists of two elements, the generic point $\zeta$ corresponding to the zero ideal and the closed point $(x)$. Define an $\mathcal{O}_X$-module $\mathcal{F}$ by setting $\mathcal{F}(X)  = \{0\}$ and $\mathcal{F}(\zeta) = k(x).$ Now $\mathcal{F}$ is not a quasi-coherent sheaf because if $\mathcal{F}|_{\operatorname{Spec} k[x]_{(x)}} = \mathcal{F}$ is isomorphic to $\widetilde{M}$ for some $A$-module $M$, $\mathcal{F}(X) = 0$ implies that $\widetilde{M}(X) = M = 0$. But now $\mathcal{F}(\zeta)$ cannot be isomorphic to $\widetilde{M}(\zeta)$ because one is non-zero while the other is zero. Thus $\mathcal{F} \notin \operatorname{QCoh}(X)$.
Are there any other examples of $\mathcal{O}_X$-modules that are not quasi-coherent sheaves?

REPLY [17 votes]: Since @Benja asks, here is an example of a non-quasi-coherent sheaf $\mathcal I$ on   the spectrum of a ring which is not a discrete valuation ring, namely $X=Spec (k[T])=\mathbb A^1_k$ ($k$ a field):  
Consider the origin $O\in X=\mathbb A^1_k$ correponding to the maximal ideal $(T)\subset k[T]$ and define the ideal  subsheaf  $\mathcal I(U)\subset \mathcal O_X(U)$  by: 
$\begin{cases} \mathcal I(U)= \mathcal O_X(U)\;\text {if}\; O\notin U      \\
  \mathcal I(U)= 0\;\text {if} \; O\in U                                 \end{cases}                     $
The sheaf $\mathcal I$ is  not quasi-coherent because  $\mathcal I\neq 0$ although  $\mathcal I(X)=0$.