We use the Shreve hydraulic potential equation as a simplified approach to investigate potential subglacial lake locations and meltwater drainage pathways beneath the Antarctic and Greenland ice sheets. We validate the method by demonstrating its ability to recall the locations of >60% of the known subglacial lakes beneath the Antarctic Ice Sheet. This is despite uncertainty in the ice-sheet bed elevation and our simplified modelling approach. However, we predict many more lakes than are observed. Hence we suggest that thousands of subglacial lakes remain to be found. Applying our technique to the Greenland Ice Sheet, where very few subglacial lakes have so far been observed, recalls 1607 potential lake locations, covering 1.2% of the bed. Our results will therefore provide suitable targets for geophysical surveys aimed at identifying lakes beneath Greenland. We also apply the technique to modelled past ice-sheet configurations and find that during deglaciation both ice sheets likely had more subglacial lakes at their beds. These lakes, inherited from past ice-sheet configurations, would not form under current surface conditions, but are able to persist, suggesting a retreating ice-sheet will have many more subglacial lakes than advancing ones. We also investigate subglacial drainage pathways of the present-day and former Greenland and Antarctic ice sheets. Key sectors of the ice sheets, such as the Siple Coast (Antarctica) and NE Greenland Ice Stream system, are suggested to have been susceptible to subglacial drainage switching. We discuss how our results impact our understanding of meltwater drainage, basal lubrication and ice-stream formation.