The room-temperature Raman signatures from graphene layers on sapphire and glass substrates were compared with those from graphene on GaAs substrate and on the standard Si/SiO2 substrate, which served as a reference. It was found that while G peak of graphene on Si/SiO2 and GaAs is positioned at 1580 cm-1 it is down-shifted by ~5 cm-1 for graphene-on-sapphire (GOS) and, in many cases, splits into doublets for graphene-on-glass (GOG) with the central frequency around 1580 cm-1. The obtained results are important for graphene characterization and its proposed graphene applications in electronic devices.