Accurate and dynamic mapping of water and ice surfaces is directly useful to navigation and lake ice cover monitoring to study climate change. Water and ice maps are also useful for various scientific applications such as atmospheric correction of satellite imagery, remote sensing of water quality, and as input data for hydrological, weather and climate models. The existing literature shows that multi-spectral satellite imagery, as provided by Sentinel-2 and Landsat-8, provides a very effective means to discriminate between water, land, and ice. However, most studies focus either on very specific cases (a specific lake for instance), or on general cases but without complex and yet very frequent cases such as turbid waters and salt lakes which can be confused with snow and ice. The Copernicus High-Resolution Snow and Ice Monitoring Service provides an operational Sentinel-2 ice and water classification product at 20m resolution but with a lot of confusion on the aforementioned cases. Using a database of 31 fully hand-labelled Sentinel-2 L2A atmospherically corrected images, and machine learning SVM and RandomForest methods, the current study shows that the classification of land, water, ice, snow, turbid waters, salt lake categories can be achieved with an accuracy over 93%. It is also shown that the atmospheric correction has little to no impact on the results, as training and evaluating from L1C top of the atmosphere images instead of L2A images yields very similar results. This last find is very useful as it means that very accurate surface masks can now be provided to atmospheric processors and may therefore considerably improve the quality of atmospherically corrected images when compared to the current usage of static masks.