<p>Security of supply is a common and important concern when integrating renewables in net-zero power systems. Extreme weather affects both demand and supply, leading to power system stress; in Europe this stress spreads continentally beyond the meteorological root cause. Here we use an approach based on shadow prices to identify periods of elevated stress called system-defining events and analyse their impact on the power system. By classifying different types of system-defining events, we identify challenges to power system operation and planning. Crucially, we find the need for sufficient resilience back-up (power) capacities whose financial viability is precarious owing to weather variability and weather-induced risk. Furthermore, we disentangle short- and long-term resilience challenges (from multi-day to annual scale) with distinct metrics and stress tests to incorporate both into future energy modelling assessments. Our methodology and implementation in an open energy system model (PyPSA-Eur) can be re-applied to other systems and help researchers and policymakers in building more resilient and adequate energy systems.</p>