The rate-dependent mechanical response of snow is commonly believed to directly inherit the ductile-to-brittle transition (DBT) from the viscoplastic ice matrix. Recent work has however stressed the impact of microstructure evolution by rapid sintering during deformation. To understand these phenomena we have conducted deformation-controlled compression experiments in an X-ray tomography stage. By varying the strain rate over 3 orders of magnitude, we find an intermediate regime where the stress response changes from smooth to serrated behavior. This regime is accompanied by microstructure quakes associated with strain localization bands in the sample interior. To interpret the results we developed a minimal, scalar model with a rate-dependent, elastoplastic constitutive law and healing which falls into the general class of rate-and-state models. The model correctly predicts the range of the instability as well as amplitude and frequency of the serrations and reveals a formal equivalence of the observed (compressive) stick-slip with seismic faults.