Seismic J waves, shear waves that traverse Earth's inner core, provide direct constraints on the inner core's solidity and shear properties. However, these waves have been elusive in the direct seismic wavefield because of their small amplitudes. We devised a new method to detect J waves in the earthquake coda correlation wavefield. They manifest through the similarity with other compressional core-sensitive signals. The inner core is solid, but relatively soft, with shear-wave speeds and shear moduli of 3.42 +/- 0.02 kilometers per second and 149.0 +/- 1.6 gigapascals (GPa) near the inner core boundary and 3.58 +/- 0.02 kilometers per second and 167.4 +/- 1.6 GPa in Earth's center. The values are 2.5% lower than the widely used Preliminary Earth Reference Model. This provides new constraints on the dynamical interpretation of Earth's inner core.