We develop new techniques to study regularity questions for moduli spaces of pseudoholomorphic curves that are multiply covered. Among the main results, we show that unbranched multiple covers of closed holomorphic curves are generically regular, and simple index $0$ curves in dimensions greater than four are generically super-rigid, implying, e.g., that the Gromov-Witten invariants of Calabi-Yau $3$-folds reduce to sums of local invariants for finite sets of embedded curves. We also establish partial results on super-rigidity in dimension four and regularity of branched covers, and briefly discuss the outlook for bifurcation analysis. The proofs are based on a general stratification result for moduli spaces of multiple covers, framed in terms of a representation-theoretic splitting of Cauchy-Riemann operators with symmetries.