Lead halide perovskite nanocrystals (NCs) have outstanding photoluminescence (PL) properties and excellent potential for light-emitting diodes and single-photon sources. Here, we report the multiple-peak structures originating from excitons, trions, and biexcitons in low-temperature PL spectra of single CsPbBr NCs. We found fine-structure splitting in the PL peaks of bright excitons and biexcitons and also in the longitudinal-optical (LO)-phonon replicas of excitons. LO-phonon replicas of trions are clearly observed under strong photoexcitation, which do not show fine-structure splitting. From size-dependent analyses of these replicas, we clarified that both exciton-phonon and trion-phonon couplings become larger for smaller NCs and the coupling strengths of trions are larger than those of excitons in large NCs. These behaviors can be explained by the spatial distributions of the electron and hole wave functions in the NCs. Our findings provide essential information on electron-phonon couplings in perovskites and for the design of high-purity single-photon sources.