  We consider effectively one-dimensional planar and radial kinks in
two-dimensional nonlinear Klein-Gordon models and focus on the sine-Gordon
model and the $\phi^4$ variants thereof. We adapt an adiabatic invariant
formulation recently developed for nonlinear Schr{\"o}dinger equations, and we
study the transverse stability of these kinks. This enables us to characterize
one-dimensional planar kinks as solitonic filaments, whose stationary states
and corresponding spectral stability can be characterized not only in the
homogeneous case, but also in the presence of external potentials. Beyond that,
the full nonlinear (transverse) dynamics of such filaments are described using
the reduced, one-dimensional, adiabatic invariant formulation. For radial
kinks, this approach confirms their azimuthal stability. It also predicts the
possibility of creating stationary and stable ring-like kinks. In all cases we
corroborate the results of our methodology with full numerics on the original
sine-Gordon and $\phi^4$ models.
