More than a decade ago, the IceCube Neutrino Observatory discovered a diffuse flux of 10 TeV–10 PeV neutrinos from our Universe. This flux of unknown origin most likely emanates from an extragalactic population of neutrino sources, which are individually too faint to appear as bright emitters. We review constraints on extragalactic neutrino source populations based on the nondetection of the brightest neutrino source. Extending previous work, we discuss limitations of source populations based on general neutrino luminosity functions. Our method provides more conservative but also statistically more robust predictions for the expected number of observable sources. We also show that the combined search of the brightest neutrino sources via weighted stacking searches or the analysis of non-Poissonian fluctuations in event-count histograms can improve the discovery potential by a factor of 2–3 relative to the brightest source.

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• Cosmic ray sources
• Cosmic rays & astroparticles
• Extrasolar neutrino astronomy