  We present two methods for characterization of motional-mode configurations
that are generally applicable to the weak and strong-binding limit of single or
multiple trapped atomic ions. Our methods are essential to realize control of
the individual as well as the common motional degrees of freedom. In
particular, when implementing scalable radio-frequency trap architectures with
decreasing ion-electrode distances, local curvatures of electric potentials
need to be measured and adjusted precisely, e.g., to tune phonon tunneling and
control effective spin-spin interaction. We demonstrate both methods using
single $^{25}$Mg$^+$ ions that are individually confined $40\,\mu$m above a
surface-electrode trap array and prepared close to the ground state of motion
in three dimensions.
