The first supernovae will soon be visible at the edge of the observable universe, revealing the birthplaces of Population III stars. With upcoming near-infrared missions, a broad analysis of the detectability of high-$z$ supernovae is paramount. We combine cosmological and radiation transport simulations, instrument specifications, and survey strategies to create synthetic observations of primeval core-collapse, Type IIn and pair-instability supernovae with the James Webb Space Telescope ($JWST$). We show that a dedicated observational campaign with the $JWST$ can detect up to $sim 15$ pair-instability explosions, $sim 300$ core-collapse supernovae, but less than one Type IIn explosion per year, depending on the Population III star formation history. Our synthetic survey also shows that $approx 1-2 times10^2$ supernovae detections, depending on the accuracy of the classification, are sufficient to discriminate between a Salpeter and flat mass distribution for high redshift stars with a confidence level greater than 99.5 per cent. We discuss how the purity of the sample affects our results and how supervised learning methods may help to discriminate between CC and PI SNe.
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