We describe spontaneous parametric down-conversion of a single-photon pump in quadratic nonlinear waveguides and waveguide arrays, taking into account spectral broadening of the signal and idler photons. We perform a detailed analysis of the photon-pair intensities, spectra and spatial correlations for different types of phase-matching conditions and identify suppression of Rabi-like oscillations due to spectral dispersion. We also discuss distinct features of signal and idler photon correlations related to the single-photon nature of the pump. © 2014 Elsevier B.V. All rights reserved.