Let $K$ be a local field and fix an algebraic closure $\bar{K}$. Local class field theory says basically that $L\mapsto N_{L/K}(L^\times)$ is a order-reversing bijection between the finite abelian extensions of $K$ contained in $\bar{K}$ and the subgroups of $K^\times$ with finite index.
Possibly, the simplest case is that of $K=\mathbb{Q}_p$, for $p>2$. I've calculated that $$\mathbb{Q}_p^\times \cong \mathbb{Z}_p\times \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}$$ and so it should be possible to determine all its finite-index subgroups and possibly classify the finite abelian extensions of $\mathbb{Q}_p$, even if we can't describe those extensions explicitly (which I presume is possible using Lubin-Tate theory, but that's not exactly the point of my question).
How could we apply such a procedure?
$$N(\Bbb{Q}_p(\zeta_{p^n}\zeta_{p^f-1})^\times)=p^{f\Bbb{Z}} (\Bbb{Z}_p^{\times})^{p^{n-1}(p-1)}$$ Then send $p^r a\in p^\Bbb{Z}\Bbb{Z}_p^\times$ to the automorphism $$\zeta_{p^n}\zeta_{p^f-1}\to \zeta_{p^n}^a\zeta_{p^f-1}^{p^r}$$ This is your isomorphism $$\Bbb{Q}_p^\times/N(\Bbb{Q}_p(\zeta_{p^n}\zeta_{p^f-1})^\times)\to Gal(\Bbb{Q}_p(\zeta_{p^n}\zeta_{p^f-1})/\Bbb{Q}_p)$$ For each subgroup $H'$ of the LHS you get a subgroup $H$ of the RHS and an abelian extension $$K_H=\Bbb{Q}_p(\zeta_{p^n}\zeta_{p^f-1})^H=\Bbb{Q}_p(\{\sum_{\sigma \in H} \sigma((\zeta_{p^n}\zeta_{p^f-1}))^j),j<p^{n-1}(p-1)f\})$$ with norm group $H'$.