Accurate and extensive bedrock geometry data is essential in ice sheet modelling. The shape of the bedrock on fine scales can influence ice sheet evolution, for example through the formation of pinning points that alter grounding line dynamics. Here we test the sensitivity of the BISICLES adaptive mesh ice sheet model to small-amplitude height fluctuations on different spatial scales in the bedrock topography provided by Bedmap2 in the catchments of Pine Island Glacier, the Amery Ice shelf and a region of East Antarctica including the Aurora Basin, Law Dome and Totten Glacier. We generate an ensemble of bedrock topographies by adding random noise to the Bedmap2 data with amplitude determined by the accompanying estimates of bedrock uncertainty. We find that the small-amplitude fluctuations result in only minor changes in the way these glaciers evolve. However, lower-frequency noise, with a broad spatial scale (over tens of kilometres) is more important than higher-frequency noise even when the features have the same height amplitudes and the total noise power is maintained. This is cause for optimism regarding credible sea level rise estimates with presently achievable density of thickness measurements. Pine Island Glacier and the region around Totten Glacier and Law Dome undergo substantial retreat and appear to be more sensitive to errors in bed topography than the Amery Ice shelf region which remains stable under the present-day observational data uncertainty.