We present calculations of the stochastic gravitational wave background resulting from neutron star birth throughout the Universe, including order-of-magnitude estimates for post-collapse bar-mode instabilities based on simulations by Brown (2000 Phys. Rev. D 62 084024) and Shibata et al (2002 Mon. Not. R. Astron. Soc. 334 L27). We employ three waveforms from Dimmelmeier et al (2002 Astron. Astrophys. 393 523) based on models, incorporating general relativistic effects, for the axisymmetric core collapse of rotating massive stars. Source-rate evolution is accounted for by using a star formation rate simulation based on a 'flat-Lambda' cosmology by Hernquist and Springel (2003 Mon. Not. R. Astron. Soc. 341 1253). We find that the core-collapse background signal is not detectable by cross correlating two advanced LIGO detectors, but a background generated by bar-mode instabilities is potentially detectable in one year of integration time.