In this work, on the basis of the previously developed continuous moment-membrane theory for describing deformations of two-dimensional nanomaterials (including graphene), three applied problems are considered for a rectangular graphene sheet: (1) the problem of static transverse bending, (2) the problem of stability of its initial compressed state, (3) the problem of its natural transverse bending vibrations. For the problem of static transverse bending of a graphene sheet, the functions of displacement and free rotations, internal force and moment characteristics are determined. In the case of the problem of the stability of the initially compressed state of a graphene sheet, the critical load is determined. When studying the problem of its natural vibrations, the frequencies of natural transverse-bending vibrations are determined, showing that the lowest natural vibration frequency of a graphene sheet is in giga- terahertz (GHz, THz) frequency region.