In this article, we explore the connections between nonnegativity, the theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semi-algebraic sets that are parametrized by families of tropical hypersurfaces. As an application, we characterization generic support sets for which the sonc-cone is equal to the sparse nonnegativity cone, and we describe a semi-algebraic stratification of the boundary of the sonc-cone in the univariate case.