Any problem in gauge theory can of course in principle be solved in any gauge (or, in the case of classical gauge theory, without using gauge potentials at all), but some gauges are much more useful than others. In gauge quantum field theory, there are a variety of gauges that are particularly useful in various situations: e.g. Coulomb gauge in the canonical formalism of QED, and various $R_\xi$ gauges (Landau, Feynman-'t Hooft, Yennie, etc.) in nonabelian gauge theory.
But in classical EM, I don't know of any situation in which any gauge other than Lorenz gauge is more practically useful than Lorenz gauge for calculations. By far the second-most widely used gauge after Lorenz gauge is Coloumb gauge. I've seen it used in two situations in classical EM, but in neither do I think it's more useful than Lorenz gauge:
In electrostatic and/or magnetostatic situations. But in these situations Coulomb and Lorenz gauges are actually equivalent, because you don't need to worry about the retardation effects that distinguish them in general. So while it's conceptually simpler to use the Coulomb gauge-fixing condition, I don't really consider it to be "better" than Lorenz gauge since it results in the exact same gauge fields.
I've seen an exercise in Jackson that uses Coulomb gauge in an electrodynamic situation in order to explicitly demonstrate that the apparently acausal behavior of Coulomb gauge actually leads to completely causal behavior in all the gauge-invariant quantities. While this is certainly a very useful pedagogical exercise, I remember the calculation being a bit messy, and I'm not sure if it would ever be the most efficient way to actually get a solution.
(I know that Lorenz gauge in incomplete and that further gauge-fixing can sometimes be useful, but I'm not counting that as a separate gauge.)
Are than any situations in classical E&M in which a gauge other than Lorenz gauge is strictly more convenient than Lorenz gauge?