  Ion Coulomb crystals are currently establishing themselves as a highly
controllable test-bed for mesoscopic systems of statistical mechanics. The
detailed experimental interrogation of the dynamics of these crystals however
remains an experimental challenge. In this work, we show how to extend the
concepts of multi-dimensional nonlinear spectroscopy to the study of the
dynamics of ion Coulomb crystals. The scheme we present can be realized with
state-of-the-art technology and gives direct access to the dynamics, revealing
nonlinear couplings even in the presence of thermal excitations. We illustrate
the advantages of our proposal showing how two-dimensional spectroscopy can be
used to detect signatures of a structural phase transition of the ion crystal,
as well as resonant energy exchange between modes. Furthermore, we demonstrate
in these examples how different decoherence mechanisms can be identified.
