The instrument package SEIS (Seismic Experiment for Internal Structure) with the three very broadband and three short-period seismic sensors is installed on the surface on Mars as part of NASA’s InSight Discovery mission. When compared to terrestrial installations, SEIS is deployed in a very harsh wind and temperature environment that leads to inevitable degradation of the quality of the recorded data. One ubiquitous artifact in the raw data is an abundance of transient one-sided pulses often accompanied by high-frequency spikes. These pulses, which we term “glitches”, can be modeled as the response of the instrument to a step in acceleration, while the spikes can be modeled as the response to a simultaneous step in displacement. We attribute the glitches primarily to SEIS-internal stress relaxations caused by the large temperature variations to which the instrument is exposed during a Martian day. Only a small fraction of glitches correspond to a motion of the SEIS package as a whole caused by minuscule tilts of either the instrument or the ground. In this study, we focus on the analysis of the glitch+spike phenomenon and present how these signals can be automatically detected and removed from SEIS’ raw data. As glitches affect many standard seismological analysis methods such as receiver functions, spectral decomposition and source inversions, we anticipate that studies of the Martian seismicity as well as studies of Mars’ internal structure should benefit from deglitched seismic data.