  The latest improvements in the scale and calibration of Type Ia supernovae
catalogues allow us to constrain the specific nature and evolution of dark
energy through its effect on the expansion history of the universe. We present
the results of Bayesian cosmological model comparison on the SNe~Ia catalogue
Pantheon+, where Flat $\Lambda$CDM is preferred by the data over all other
models and we find moderate evidence ($\Delta \log \mathcal{Z} \sim 2.5$) to
reject a number of the alternate dark energy models. The effect of peculiar
velocity corrections on model comparison is analysed, where we show that
removing the peculiar velocity corrections results in a varying fit on
non-$\Lambda$CDM parameters. As well as comparing cosmological models, the
Bayesian methodology is extended to comparing the scatter model of the data,
testing for non-gaussianity in the Pantheon+ Hubble residuals. We find that
adding a scale parameter to the Pantheon+ covariances, or alternately using a
multivariate Student's t-distribution fits the data better than the fiducial
analysis, producing a cosmology independent evidence increase of $\Delta \log
\mathcal{Z} = 2.29 $ and $2.46$ respectively. This improved treatment of the
scatter decreases the uncertainty in the constraint on the Hubble constant,
finding $H_0 = 73.67 \pm 0.99 $ km s$^{-1}$ Mpc$^{-1}$, in $ 5.7 \sigma$
tension with Planck. We also explore $M_B$ transition models as a potential
solution for the Hubble tension, finding no evidence to support these models
among the SNe data.
