We present a fast and efficient method for studying vacuum stability constraints in multi-scalar theories beyond the Standard Model. This method is designed for a reliable use in large scale parameter scans. The minimization of the scalar potential is done with the well-known polynomial homotopy continuation, and the decay rate of a false vacuum in a multi-scalar theory is estimated by an exact solution of the bounce action in the one-field case. We compare to more precise calculations of the tunnelling path at the tree- and one-loop level and find good agreement for the resulting constraints on the parameter space. Numerical stability, runtime and reliability are significantly improved compared to approaches existing in the literature. This procedure is applied to several phenomenologically interesting benchmark scenarios defined in the Minimal Supersymmetric Standard Model. We utilize our efficient approach to study the impact of simultaneously varying multiple fields and illustrate the importance of correctly identifying the most dangerous minimum among the minima that are deeper than the electroweak vacuum.