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We explore the properties of the density matrix of a random quantum error-correcting code initialized in a particular logical state and subject to single-qubit noise. The entanglement structure of this density matrix changes abruptly at the error correction threshold; the error correction threshold thus exemplifies a “mixed-state phase transition.” Mixed-state phase transitions are invisible to simple observables that are linear in the density matrix (e.g., expectation values or correlation functions), but correspond to singularities in nonlinear observables like the entropy and the coherent information. We use random-matrix theory to characterize how the spectrum of the decohered density matrix evolves with noise. At low error rates, the spectrum of the density matrix consists of well-separated “bands” of eigenvalues corresponding to errors of different weights. The level statistics within each band follows a Marchenko-Pastur distribution. Since there are parametrically more possible errors than eigenstates, the eigenvalue-error correspondence breaks down for very small eigenvalues of the density matrix: these correspond to uncorrectable errors, which are possible but exponentially unlikely below threshold. As the noise strength is increased, the probability of uncorrectable errors increases, and a spectral transition (the error correction threshold) takes place. For a random code this threshold occurs at the quantum hashing bound. Past the error correction threshold, typical errors are uncorrectable. However, low-weight errors (corresponding to the largest-amplitude eigenvalues of the density matrix) remain correctable: i.e., error correction remains possible if one post-selects on seeing certain error syndromes. We calculate a line of post-selected error correction thresholds, interpolating between the standard error-correction threshold and the error detection threshold, as well as a separate line of transitions.