For a long time, I had a false belief that the space/stack $\text{Coh}^{tf}_{c_1,c_2}S$ of torsion-free sheaves $\mathcal{E}$ on a smooth algebraic surface $S$ was not connected, since if you take its map into its double dual $$0\ \to \ \mathcal{E}\ \to\ \mathcal{E}^{\vee\vee}\ \to \ \mathcal{Q}\ \to\ 0$$ the length of the torsion sheaf $\mathcal{Q}$ gives another invariant labelling connected components.

What are some "typical" examples of (say, flat one parameter) families of torsion-free sheaves where the torsion quotient jumps in length?