  Caustic singularity formations in shift-symmetric $k$-essence and Horndeski
theories on a fixed Minkowski spacetime were recently argued. In $n$
dimensions, this singularity is the $(n-2)$-dimensional plane in spacetime at
which second derivatives of a field diverge and the field loses single-valued
description for its evolution. This does not necessarily imply a pathological
behavior of the system but rather invalidates the effective description. The
effective theory would thus have to be replaced by another to describe the
evolution thereafter. In this paper, adopting the planar-symmetric
$1$+$1$-dimensional approach employed in the original analysis, we seek all
$k$-essence theories in which generic simple wave solutions are free from such
caustic singularities. Contrary to the previous claim, we find that not only
the standard canonical scalar but also the DBI scalar are free from caustics,
as far as planar-symmetric simple wave solutions are concerned. Addition of
shift-symmetric Horndeski terms does not change the conclusion.
