We present temporally evolving 3‐D global mantle convection models with single‐sided subduction and a free surface in both 3‐D Cartesian and fully spherical geometry. Special focus is given to the spontaneous development of three‐dimensional structures at the surface and in the upper mantle. We find that an arcuate shape is the natural form for trenches and slabs. Cartesian models are used first to study the dynamic evolution of subduction zones, spreading ridges, and interconnected transform features. These experiments highlight the strong variation of spontaneously developing, arcuate slab curvature and subduction polarity along the trench strike. The spontaneous development of spreading ridges leads to lateral offsets between separated segments that are characterized by normal transform motion. Spherical models then allow insights into the evolution of plate tectonics on a sphere. Investigated are the spontaneous evolution of slab geometry, trench motion, and subduction‐induced mantle flow. Two new dynamical features are discovered: “back‐slab spiral flow” and “slab tunneling.” © 2014. American Geophysical Union. All Rights Reserved.