We show the existence and investigate the dynamics and statistics of rogue oscillations (standing waves) generated in the frame of the nonlinear quantum harmonic oscillator (NQHO), also known as the Gross-Pitaevskii equation (GPE). With this motivation, in this paper we develop a split-step Fourier scheme for the computational analysis of NQHO. We show that modulation instability excites the generation of rogue oscillations in the frame of the NQHO. We also discuss the effects of various parameters such as the strength of trapping well potential, nonlinearity, dissipation, fundamental wave number and perturbation amplitude on rogue oscillation formation probabilities. (C) 2020 Elsevier B.V. All rights reserved.