  Using CORSIKA simulations, we investigate the mass sensitivity of cosmic-ray
air-shower observables for sites at the South Pole and Malarg\"ue, Argentina,
the respective locations of the IceCube Neutrino Observatory and the Pierre
Auger Observatory. Exact knowledge of observables from air-shower simulations
was used to study the event-by-event mass separation between proton, helium,
oxygen, and iron primary cosmic rays with a Fisher linear discriminant
analysis. Dependencies on the observation site as well as the energy and zenith
angle of the primary particle were studied in the ranges from
$10^{16.0}-10^{18.5}\,$eV and $0^\circ$ to $60^\circ$: they are mostly weak and
do not change the qualitative results. Promising proton-iron mass separation is
achieved using combined knowledge of all studied observables, also when typical
reconstruction uncertainties are accounted for. However, even with exact
measurements, event-by-event separation of intermediate-mass nuclei is
challenging and better methods than the Fisher discriminant and/or the
inclusion of additional observables will be needed. As an individual
observable, high-energy muons ($> 500\,$GeV) provide the best event-by-event
mass discrimination, but the combination of muons of any energy and
$X_{\text{max}}$ provides already a high event-by-event separation between
proton-iron primaries at both sites. We also confirm that the asymmetry and
width parameters of the air-shower longitudinal profile, $R$ and $L$, are mass
sensitive. Only $R$ seems to be suitable for event-by-event mass separation,
but $L$ can potentially be used to statistically determine the proton-helium
ratio. Overall, our results motivate the coincident measurement of several
air-shower observables, including at least $X_{\text{max}}$ and the sizes of
the muonic and electromagnetic shower components, for the next generation of
air-shower experiments.
