Sills, saucer-shaped sills, and cone sheets are fundamental magma conduits in many sedimentary basins worldwide. Models of their emplacement usually approximate the host rock properties as purely elastic and consider the plastic deformation to be negligible. However, many field observations suggest that inelastic damage and shear fracturing play a significant role during sill emplacement. Here we use a rigid plasticity approach, through limit analysis modeling, to study the conditions required for inelastic deformation of sill overburdens. Our models produce distinct shear failure structures that resemble intrusive bodies, such as cone sheets and saucer-shaped sills. This suggests that shear damage greatly controls the transition from flat sill to inclined sheets. We derive an empirical scaling law of the critical overpressure required for shear failure of the sill's overburden. This scaling law allows to predict the critical sill diameter at which shear failure of the overburden occurs, which matches the diameters of natural saucer-shaped intrusions' inner sills. A quantitative comparison between our shear failure model and the established sill's tensile propagation mechanism suggests that sills initially propagate as tensile fractures, until reaching a critical diameter at which shear failure of the overburden controls the subsequent emplacement of the magma. This comparison also allows us to predict, for the first time, the conditions of emplacement of both conical intrusions, saucer-shaped intrusions, and large concordant sills. Beyond the application to sills, our study suggests that shear failure significantly controls the emplacement of igneous sheet intrusions in the Earth's brittle crust. ©2018. American Geophysical Union. All Rights Reserved.