  We study locally conformally balanced metrics on almost abelian Lie algebras,
namely solvable Lie algebras admitting an abelian ideal of codimension one,
providing characterizations in every dimension. Moreover, we classify
six-dimensional almost abelian Lie algebras admitting locally conformally
balanced metrics and study some compatibility results between different types
of special Hermitian metrics on almost abelian Lie groups and their compact
quotients. We end by classifying almost abelian Lie algebras admitting locally
conformally hyperk\"ahler structures.
