The first part of this paper treats the axisymmetric transverse vibrations of membrane shells. The stress differential equations of motion with longitudinal inertia neglected are solved for a class of shells that includes the ogive, cone, cylinder, and sphere. The emphasis is placed on predicting the lowest natural frequency of vibration for shells with a free boundary. These frequencies are compared with the previously known results of the bending solutions for the shallow spherical‐shell segment and for the hemispherical shell. In the second part, attention is turned to the shell whose mode of deformation is primarily flexural by applying hypotheses associated with edge effects. The deformation is specified by a pair of differential equations that are solved for a class of shells of revolution. The frequency equation is obtained and particularized to specific geometries.