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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.