In this talk we consider the Euler point vortices system, and identify a condition under which self-similar bursts and collapses of three isolated vortices in the plane are stable with respect to a sufficiently regular external perturbation. The condition is related to the stability of the linearized dynamics around three vortices evolving in a self-similar fashion. As a consequence, we show existence of bursts out of (and collapses into) any given initial configuration of vortices, both in the plane and in bounded domains. Finally, we discuss extensions to the generalized Surface Quasi-Geostrophic point vortices system. Joint work with Francesco Grotto.