Pre-print, Working paper: The aim of this paper is to study the performance of residual-based tests for cointegration in the presence of multiple deterministic structural breaks via Monte Carlo simulations. We consider the KPSS-type LM tests proposed in Carrion-i-Silvestre and Sansò (2006) and in Bartley, Lee and Strazicich (2001), as well as the Schmidt and Phillips-type LM tests proposed in Westerlund and Edgerton (2007). This exercise allow us to cover a wide set of single-equation cointegration estimators. Monte Carlo experiments reveal a trade-off between size and power distortions across tests and models. KPSS-type tests display large size distortions under multiple breaks scenarios, while Schmidt and Phillips-type tests appear well-sized across all simulations. However, when regressors are endogenous, the former group of tests displays quite high power against the alternative hypothesis, while the latter shows severe low power.