In this paper, the performance of optical spatial modulation (OSM) in the indoor multipath optical wireless channel is investigated. Multipath propagation of the transmitted signal results in the temporal dispersion which causes intersymbol interference (ISI) especially in high-speed communications. OSM schemes have been investigated in line-of-sight (LOS) channels. However, given the recent trend in Gigabits per second communication, there is need to investigate the impact of the neglected higher-order reflections. Two variants of OSM are explored as case studies: optical space shift keying (OSSK) and spatial pulse position modulation (SPPM). The multipath-induced ISI is modelled to account for the spreading of the transmitted signal, and the analytical upper bound on the symbol error rate of both OSM schemes in multipath channel is derived. The derived analytical bounds are validated by closely matching simulation results. Furthermore, using the spatial distributions of the multipath-induced power penalty, the LOS channel response and the delay spread across the room, we demonstrate how the interaction between these parameters impacts error performance. Multipath-induced ISI has a significant adverse impact on the performance of OSM schemes, particularly on the detection of the activated transmitter as the incurred ISI can alter the channel gain of the received symbol.