  We study singularities in tropical hypersurfaces defined by a valuation over
a field of positive characteristic. We provide a method to compute the set of
singular points of a tropical hypersurface in positive characteristic and the
p-adic case. This computation is applied to determine all maximal cones of the
tropical linear space of univariate polynomials of degree $n$ and
characteristic $p$ with a fixed double root and the fan of all tropical
polynomials that have $0$ as a double root independently of the characteristic.
We also compute, by pure tropical means, the number of vertices, edges and
2-faces of the Newton polytope of the discriminant of polynomials of degree $p$
in characteristic $p$.
