  The concept of wave front set was introduced in 1969-1970 by M. Sato in the
hyperfunctions context and by L. H\"ormander in the $\mathcal C^{\infty}$
context. Howe used the theory of wave front sets in the study of Lie groups
representations. Heifetz defined a notion of wave front set for distributions
in the $p$-adic setting and used it to study some representations of $p$-adic
Lie groups.
  In this article, we work in the $k((t))$-setting with $k$ a characteristic
zero field. In that setting, balls are no longer compact but working in a
definable context provides good substitutes for finiteness and compactness
properties. We develop a notion of definable distributions in the motivic
integration framework of Cluckers--Loeser for which we define notions of
singular support and $\Lambda$-wave front sets (relative to some multiplicative
subgroups $\Lambda$ of the valued field) and we investigate their behaviour
under natural operations like pull-back, tensor product, and products of
distributions.

Algebraic Geometry;Logic;Representation Theory