I think a good illustration of why torsion-free sheaves on singular curves are both interesting and difficult is given by the following. Consider the $GL_n$ case of the Hitchin fibration, i.e., the map from the moduli space of vector bundles of rank $n$ with a twisted endomorphism on a smooth, projective curve to the Hitchin base space of characteristic polynomials.

Then a result of Beauville, Naramsihan, and Ramanan (see this paper http://math.unice.fr/~beauvill/pubs/bnr.pdf) says that for a sufficiently nice characteristic polynomial $a$ in the Hitchin base, the stack of torsion-free coherent sheaves of rank one on the associated spectral curve is isomorphic to the Hitchin fiber associated to $a$. See, for example, the notes on the Hitchin fibration on Drinfeld's geometric Langlands page for a quick introduction to these ideas.

In general, these spectral curves will be singular (which is why I couldn't simply say 'line bundle' in the above correspondence). Given that the Hitchin fibration and Hitchin fibers are some of the most interesting geometric objects currently being studied, I think this gives a flavor for how interesting torsion-free sheaves on singular curves (and these are just rank one) can be.

Also, it's worth mentioning that the curves which arise as spectral curves aren't even that singular (nodal and cuspidal elliptic curves are a couple examples), in the sense that the dimension of the tangent space at any point is at most two. There's an old result (I think from 1979) of Altman, Iarrobino, and Kleiman proving that in this situation, the stack of line bundles is dense in the stack of torsion-free coherent sheaves of rank one. This result has since been generalized to arbitrary reductive groups by Ngo in his paper proving the Fundamental Lemma.