Example of a module which is free at an isolated point I'm looking for the most simple example of a quasicoherent sheaf $\mathcal{F}$ over a scheme $X$ (preferably affine for simplicity) which has a free stalk $\mathcal{F}_x$ at a point $x \in X$ and yet for every open neighborhood $x \in U$ the restricted sheaf $\mathcal{F}|_{U}$ isn't free. 
By semicontinuity theorem $\mathcal{F}$ must jump down rank at $x$. I'm trying to figure out how badly behaved these jumps are.
In terms of modules I'm looking for a module $M$ over $A$ which has a free localization $M_{\mathfrak{p}}$ but s.t. for every $f \notin \mathfrak p\subset A$ the localization $M_f$ isn't free. 
Bonus points: Find such a finite module over a noetherian ring!

I'm surprised no one has mentioned the following very simple example.  Let $A=\mathbb{Z}$ and let $M=\mathbb{Q}$.  Then $M$ is free at the generic point of $\operatorname{Spec} A$, but is not free in any open neighborhood.  Or, if you want $x$ to be a closed point, you can instead take $M=\mathbb{Z}_{(p)}$ for some prime $p\in\mathbb{Z}$.
More generally, if $X$ is any scheme and $x\in X$ is any point such that every neighborhood of $x$ contains a point which is not a generalization of $x$, then the local ring at $x$ will be free at $x$ but not in any neighborhood of $x$ (it can't be free in any neighborhood because its fiber is $1$-dimensional at $x$ but $0$-dimensional at any point which is not a generalization of $x$).
