Subduction provides the main driving force for the motion of tectonic plates at the Earth’s surface through slab-pull and sinking-induced flow in the surrounding mantle. The ability of the slab to directly transmit slab-pull forces to the tectonic plate at the surface depends on the minimum strength and rheology (e.g., viscous, plastic) of the slab. Previous models have shown that observations including the state of stress in slabs, dynamic topography and the geoid above slabs, the evolution of slab and the kinematic history of subduction can be well-matched by a variety of models with either low viscosity (i.e., 100-1,000 times more viscous than the surrounding mantle) or high viscosity slabs (i.e., more than 10,000 times more viscous than the surrounding mantle). However, in many of the models in which a good match to observations is found for low viscosity slabs, the maximum slab viscosity is imposed as a cut-off value, which forces the entire slab to have a  more or less uniform viscosity independent of strain-rate or stress magnitude, rather than a plastic yielding-type rheology. We present numerical models demonstrating that when the non-Newtonian viscosity of the upper mantle and plastic yielding behavior of slabs are taken into account, the minimum yield strength that allows for continuous subduction is approximately 300-500 MPa, which leads to high viscosity slabs with some localized lower viscosity regions. A yield stress of 10-100 MPa is required to form uniformly low viscosity slabs, but these slabs detach from the subducting plate, due to localized weakening, when the slab reaches a length of 200-300 km, even when subduction is facilitated by a low viscosity shear zone and kinematically-imposed surface velocities. In contrast, detachment of higher strength slabs in fully-dynamic models only occurs when the shear zone is removed and prevents further subduction.