The nonlinear modal coupling between the vibration modes of an arch-shaped microstructure is an interesting phenomenon, which may have desirable features for numerous applications, such as vibration-based energy harvesters. This work presents an investigation into the potential nonlinear internal resonances of a microelectromechanical systems (MEMS) arch when excited by static (DC) and dynamic (AC) electric forces. The influences of initial rise and midplane stretching are considered. The cases of one-to-one and three-to-one internal resonances are studied using the method of multiple scales and the direct attack of the partial differential equation of motion. It is shown that for certain initial rises, it is possible to activate a three-to-one internal resonance between the first and third symmetric modes. Also, using an antisymmetric half-electrode actuation, a one-to-one internal resonance between the first symmetric and the second antisymmetric modes is demonstrated. These results can shed light on such interactions that are commonly found on micro and nanostructures, such as carbon nanotubes.