Semiconductor copper indium sulfide quantum dots are emerging as promising alternatives to cadmium- and lead-based chalcogenides in solar cells, luminescent solar concentrators, and deep-tissue bioimaging due to their inherently lower toxicity and outstanding photoluminescence properties. However, the nature of their emission pathways remains a subject of debate. Using low-temperature single quantum dot spectroscopy on core-shell copper indium sulfide nanocrystals, we observe two subpopulations of particles with distinct spectral features. The first class shows sharp resolution-limited emission lines that are attributed to zero-phonon recombination lines of a long-lived band-edge exciton. Such emission results from the perfect passivation of the copper indium sulfide core by the zinc sulfide shell and points to an inversion in the band-edge hole levels. The second class exhibits ultrabroad spectra regardless of the temperature, which is a signature of the extrinsic self-trapping of the hole assisted by defects in imperfectly passivated quantum dots.