We investigate the dynamical stability of interacting systems following a quantum quench protocol. In particular, we consider systems that are stable at a mean field level and can become unstable due to normal-mode fluctuations. We discover a new mechanism by which their eigenfrequencies acquire an imaginary part and become dynamically unstable. Our analysis involves the interplay between two weakly-coupled channels, such as the “charge” and “spin” of a weakly-imbalanced spinor condensate. When one channel is energetically stable and the other energetically unstable, the two modes necessarily cross and induce a dynamical instability. The resulting dynamics is characterized periodically-oscillating spontaneously-formed patterns. We describe two possible realizations of this mechanism using ultracold atoms and suggest that recent experiments may have already observed this phenomenon.