Following the increasing adoption of stablecoins, their robustness has become an ongoing concern for investors and regulators alike. This article studies the co-instability of a large panel of stablecoins by identifying their structural breaks and co-occurrences. Moreover, we use Dynamic Time Warping (DTW) and Dynamic Conditional Correlation DCC-GARCH models to assess spillover effects in the wake of four major black swan events in the crypto world. Similar to prior literature, we conclude that algorithmic stablecoins were the major shock receivers of the IRON TITAN and TERRA LUNA crashes, highlighting the importance of allowing for sufficient margins of safety in their design. We also observe that the FTX bankruptcy and the Silicon Valley Bank collapse had repercussions of a different nature and affected more categories of stablecoins, from algorithmic to fiat money-backed. We also highlight how, irrespective of the idiosyncratic or systemic nature of the shocks, well-known custodial stablecoins, as well as the most successful crypto-collateralized tokens, showed resilience during or in the immediate aftermath of these events.