We introduce a Bayesian method for the estimation of single-qubit errors in quantum devices, and use it to characterize these errors on three superconducting 27-qubit devices. We self-consistently estimate up to seven parameters of each qubit's state preparation, readout, and gate errors, analyze the stability of these errors as a function of time, and demonstrate easily implemented approaches for mitigating different errors before a quantum computation experiment. On the investigated devices we find non-negligible qubit reset errors that cannot be parametrized as a diagonal mixed state, but manifest as a coherent phase in a superposition with a small contribution from the qubit's excited state. We are able to mitigate such errors by applying prerotations on the initialized qubits, which we demonstrate with multiqubit entangled states. Our results show that Bayesian estimation can resolve small parameters - including those pertaining to quantum gate errors - with a high relative accuracy, at a lower measurement cost as compared with standard characterization approaches.