Local stiffness plays a critical role in cell function, but measuring rigidity at cellular length scales in living 3D tissues presents considerable challenges. Here we present thermoresponsive, smart material microgels that can be dispersed or injected into tissues and optically assayed to measure internal tissue stiffness over several weeks. We first develop the material design principles to measure tissue stiffness across physiological ranges, with spatial resolutions approaching that of individual cells. Using the microfabricated sensors, we demonstrate that mapping internal stiffness profiles of live multicellular spheroids at high resolutions reveal distinct architectural patterns, that vary with subtle differences in spheroid aggregation method. Finally, we determine that small sites of unexpectedly high stiffness (> 250 kPa) develop in invasive breast cancer spheroids, and in in vivo mouse model tumors as the cancer progresses towards metastatic disease. These highly focal sites of increased intratumoral stiffness likely form via active cell mechanical behavior, and suggest new possibilities for how early mechanical cues that drive cancer cells towards invasion might arise within the evolving tumor microenvironment.