This paper presents an investigation into the gravitational microlensing of quasars by stars and stellar remnants in the Milky Way. We present predictions for the all-sky microlensing optical depth, time-scale distributions and event rates for future large-area sky surveys. As expected, the total event rate increases rapidly with increasing magnitude limit, reflecting the fact that the number density of quasars is a steep function of magnitude. Surveys, such as Pan-STARRS and LSST, should be able to detect more than 10 events per year, with typical event durations of around 1 month. Since microlensing of quasar sources suffers from fewer degeneracies than lensing of Milky Way sources, they could be used as a powerful tool for recovering the mass of the lensing object in a robust, often model-independent, manner. As a consequence, for a subset of these events, it will be possible to directly ‘weigh’ the star (or stellar remnant) that is causing the lensing signal, either through higher order microlensing effects and/or high-precision astrometric observations of the lens star (using e.g. Gaia or SIM-lite). This means that such events could play a crucial role in stellar astronomy. Given the current operational timelines for Pan-STARRS and LSST, by the end of the decade, they could potentially detect up to 100 events. Although this is still too few events to place detailed constraints on Galactic models, consistency checks can be carried out and such samples could lead to exciting and unexpected discoveries.