A microelectromechanical microphone has a stationary region or another type of mechanically supported region that can mitigate or avoid mechanical instabilities in the microelectromechanical microphone. The stationary region can be formed in a diaphragm of the microelectromechanical microphone by rigidly attaching, via a rigid dielectric member, an inner portion of the diaphragm to a backplate of the microelectromechanical microphone. The rigid dielectric member can extend between the backplate and the diaphragm. In certain embodiments, the dielectric member can be hollow, forming a shell that is centrosymmetric or has another type of symmetry. In other embodiments, the dielectric member can define a core-shell structure, where an outer shell of a first dielectric material defines an inner opening filled with a second dielectric material. Multiple dielectric members can rigidly attach the diaphragm to the backplate. An extended dielectric member can rigidly attach a non-planar diaphragm to a backplate.