I think the answer is that you can take nth roots of a complex power of a line bundle - i.e. of a line bundle on a G_m gerbe (an example of a twisted sheaf), and this will account for preimages of elements in H^2(G_m) in your long exact sequence (see this related question).

As for interesting examples of $\mu_n$ gerbes, look at the moduli stack of stable $G$ bundles on an algebraic curve for $G$ with center $\mu_n$ (eg $SL_n$). It is a $\mu_n$ gerbe over the moduli space of bundles, whose nontriviality accounts for the lack of existence of a universal bundle. (It is described explicitly in many places - the ones that come to mind are King-Schofield arXiv:math/9907068, Beilinson-Drinfeld's Quantization of Hitchin Hamiltonians (Chapter 4) and Kapustin-Witten arXiv:hep-th/0604151 (Section 7).

I think the answer is that you can take nth roots of a complex power of a line bundle - i.e. of a line bundle on a G_m gerbe (an example of a twisted sheaf), and this will account for preimages of elements in H^2(G_m) in your long exact sequence (see this related question).

As for interesting examples of $\mu_n$ gerbes, look at the moduli stack of stable $G$ bundles on an algebraic curve for $G$ with center $\mu_n$ (eg $SL_n$). It is a $\mu_n$ gerbe over the moduli space of bundles, whose nontriviality accounts for the lack of existence of a universal bundle. (It is described explicitly in many places - the ones that come to mind are King-Schofield arXiv:math/9907068, Beilinson-Drinfeld's Quantization of Hitchin Hamiltonians (Chapter 4) and Kapustin-Witten arXiv:hep-th/0604151 (Section 7).