Consider the set of monic fourth-order real polynomials transformed so that the constant term is one. In the three-dimensional space of the coefficients describing this set, the domain of asymptotic stability is bounded by a surface with the Whitney umbrella singularity. The maximum of the real parts of the roots of these polynomials is globally minimized at the Swallowtail singular point of the discriminant surface of the set corresponding to a negative real root of multiplicity four. Motivated by this example, we review recent works on robust stability, abscissa optimization, heavily damped systems, dissipation-induced instabilities, and eigenvalue dynamics in order to point out some connections that appear to be not widely known.