  Expectation values of observables are routinely estimated using so-called
classical shadows$\unicode{x2014}$the outcomes of randomized bases measurements
on a repeatedly prepared quantum state. In order to trust the accuracy of
shadow estimation in practice, it is crucial to understand the behavior of the
estimators under realistic noise. In this work, we prove that any shadow
estimation protocol involving Clifford unitaries is stable under gate-dependent
noise for observables with bounded stabilizer norm$\unicode{x2014}$originally
introduced in the context of simulating Clifford circuits. For these
observables, we also show that the protocol's sample complexity is essentially
identical to the noiseless case. In contrast, we demonstrate that estimation of
`magic' observables can suffer from a bias that scales exponentially in the
system size. We further find that so-called robust shadows, aiming at
mitigating noise, can introduce a large bias in the presence of gate-dependent
noise compared to unmitigated classical shadows. Nevertheless, we guarantee the
functioning of robust shadows for a more general noise setting than in previous
works. On a technical level, we identify average noise channels that affect
shadow estimators and allow for a more fine-grained control of noise-induced
biases.
