We investigate how surface load variations around volcanoes act on shallow magma chambers. Numerical calculations are carried out in axisymmetric geometry for an elliptical chamber embedded in an elastic medium. Magma compressibility is taken into account. For variable chamber shape, size and depth, we quantify how unloading events induce magmatic pressure change as well as variation of the threshold pressure required for dyke initiation at the chamber wall. We evaluate the triggering effect of these surface events on onset of eruptions and find it depends strongly on the surface load location and the magma chamber shape. We apply this model to two active Icelandic subglacial volcanoes: Grímsvötn and Katla. The 2004 eruption of Grímsvötn was immediately preceded by a jökulhlaup, a glacial outburst flood of 0.5 km3. We show that this event may have triggered the eruption only if the system was very close to failure conditions. Katla volcano is covered by the Mýrdalsjökull ice cap. An annual cycle, with up to 6 m change in snow thickness, occurs from winter to summer. As the seasonal snow load is reduced, a pressure decrease of the same order of magnitude as the load is induced within the magma storage zone. Our model predicts that, in the case of a spherical or horizontally elongated magma chamber, eruptions are more likely when the snow cover is smallest, which appears consistent with the fact that all the last nine major historical eruptions at Katla occurred during the summer period. The model predicts an increase in Coulomb stress around the caldera, up to 7 km from its centre, during unloading periods, enough to trigger earthquakes. Stress due to snow load variations, with focusing of it in weak zones near the caldera boundary, is considered a contributing factor to seasonal seismicity observed beneath Mýrdalsjökull.