Abstract We present Euler Characteristic Surfaces as a multiscale spatio-temporal topological summary of time series data encapsulating the topology of the system at different time instants and length scales. Euler Characteristic Surfaces with an appropriate metric are used to quantify stability and locate critical changes in a dynamical system with respect to variations in a parameter, while being substantially computationally cheaper than available alternate methods such as Persistent Homology. The stability of the construction is demonstrated by a quantitative comparison with Persistent Homology, and a quantitative stability under small changes in time is established. Stability measures are calculated for two simulated disordered flow situations- one with conserved particle number and the other, not; the analytical conditions of stability are validated by the calculated measures. Real dynamical systems are examined for temporal stability and discussed. We also show that the tools of ECS and EM can capture signatures of order–disorder transitions in dynamical systems.