It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain(in paticular, Dedekind domain) are flat, please see [here][1]. My questiones are :

> - Is there a general condition under which a torsion free module is flat?
- What is the simplest example of torsion-free non-flat module?

 **edit:** Since the example in Georges's answer is the coordinate ring of a singular curve, I want to ask the same questiones for the coordinate ring $R$ of a smooth algebraic variety: 
> - Is there a general condition under which a torsion free module over $R$ is flat? 
- what is the simplest example of torsion-free non-flat module over  $R$  

  [1]: http://en.wikipedia.org/wiki/Pr%C3%BCfer_domain