The dynamics of a quantum pendulum described by the Mathieu-Schrödinger equation can express a complex picture of quantum chaos. The parameter-dependent energy spectrum of the Mathieu equation has special properties. This feature is that it contains many branching and merging points connecting energy levels with each other. The energy intervals between the branching and merging points correspond to non-degenerate states, and the energy regions outside these intervals correspond to degenerate states. This circumstance has caused the interest in finding the location of singular points. In this paper, the problem of a nonlinear oscillator under the resonant action of a periodic series of pulses is reduced to the problem of a quantum pendulum. Location of singular points of the energy spectrum is determined for it.