  This work examines the relationships between the properties (flux ratios,
full width at half-maximum velocities) of the [O I] $\lambda\lambda$6300, 6364,
[Ca II] $\lambda\lambda$7291, 7323, and the Ca II near-infrared triplet,
emission lines of a large sample of core-collapse supernovae (SNe) and Ca-rich
transients (509 spectra of 86 transients, of which 10 transients are Ca-rich
events). Line-flux ratios as a function of time were investigated with
differences identified between the transient classes, in particular the Type II
SNe were found to have distinct line-flux ratios compared to stripped-envelope
(SE) SNe. No correlation was found between the [Ca II]/[O I] flux ratios of
SE-SNe and their ejecta masses and kinetic energies (as measured from light
curve modelling), suggesting that there may be a contribution from an
additional power source in more luminous SE-SNe. We found that the mean
characteristic width of the [Ca II] emission line is less than the [O I]
emission line for all SN types, indicating that the [Ca II] emission typically
originates from deeper in the ejecta than [O I]. This is in some tension with
standard models for emission in Type II SNe. The emission line properties of
Type II SNe were also compared to theoretical models and found to favour lower
mass tracks ($M_\mathrm{ZAMS}$ $<$ 15 M$_{\odot}$), with no evidence found for
significant mixing of $^{56}$Ni into the H envelope nor Ca mixed into the O
shell. The flux ratios of some superluminous SNe were found to be similar to
those of SE-SNe when scaling to account for their longer rise times was applied
(although we caution the sample size is small).
