We demonstrate spontaneous generation of spatially-periodic supersolid-like super-lattice and stripe solitons in Rashba spin-orbit (SO) coupled spin-1 and spin-2 quasi-two-dimensional nonmagnetic Bose-Einstein condensates (BECs). The solitons in a weakly SO-coupled spin-1 BEC are circularly-symmetric of (−1, 0, +1) and (0, +1, +2) types and have inherent vorticity; the numbers in the parentheses are the winding numbers in hyper-spin components +1, 0, −1, respectively. The circularly-symmetric solitons in an SO-coupled spin-2 BEC are of types (−2, −1, 0, +1, +2) and (−1, 0, +1, +2, +3) with the former being the ground state, where the winding numbers correspond to spin components +2, +1, 0, −1, −2, respectively. For stronger SO-coupling strengths, these solitons acquire a multiring structure while preserving the winding numbers. Quasi-degenerate stripe and super-lattice solitons, besides a circularly-asymmetric soliton, also emerge as excited stationary states for stronger SO-coupling strengths in spin-1 and spin-2 BECs.