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. 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 $R$