We study the extended Jaynes-Cummings-Hubbard model on triangular cavity lattices and zigzag ladders. By using density-matrix renormalization-group methods, we observe various types of solids with different density patterns and find evidence for light supersolids, which exist in extended regions of the phase diagram of the zigzag ladder. Furthermore, we observe strong pair correlations in the supersolid phase due to the interplay between the atoms in the cavities and atom-photon interaction. By means of cluster mean-field simulations and a scaling of the cluster size extending our analysis to two-dimensional triangular lattices, we present evidence for the emergence of a light supersolid in this case also.