The driven nonlinear Duffing oscillator is a very good, and standard, example of a quantum mechanical system from which classical-like orbits can be recovered from unravelings of the master equation. In order to generate such trajectories in the phase space of this oscillator, in this paper we use the quantum jump unraveling together with a suitable application of the correspondence principle. We analyze the measured readout by considering the power spectra of photon counts produced by the quantum jumps. Here we show that localization of the wave packet from the measurement of the oscillator by the photon detector produces a concomitant structure in the power spectra of the measured output. Furthermore, we demonstrate that this spectral analysis can be used to distinguish between different modes of the underlying dynamics of the oscillator.