In this paper we explore the existing tensions in the local cosmological expansion rate, ${H}_{0}$, and amplitude of the clustering of the large-scale structure at $8{h}^{\ensuremath{-}1}\text{ }\text{ }\mathrm{Mpc}$, ${\ensuremath{\sigma}}_{8}$, as well as models that claim to alleviate these tensions. We consider seven models: evolving dark energy ($w\mathrm{CDM}$), extra radiation (${N}_{\mathrm{eff}}$), massive neutrinos, curvature, primordial magnetic fields (PMF), self-interacting neutrino models, and early dark energy (EDE). We test these models against three datasets that span the full range of measurable cosmological epochs, have significant precision, and are well-tested against systematic effects: the Planck 2018 cosmic microwave background data, the Sloan Digital Sky Survey baryon acoustic oscillation scale measurements, and the Pantheon catalog of type Ia supernovae. We use the recent SH0ES ${H}_{0}$ measurement and several measures of ${\ensuremath{\sigma}}_{8}$ (and its related parameter ${S}_{8}={\ensuremath{\sigma}}_{8}\sqrt{{\mathrm{\ensuremath{\Omega}}}_{\mathrm{m}}/0.3}$). We find that four models are above the ``strong'' threshold in Bayesian model selection, $w\mathrm{CDM}$, ${N}_{\mathrm{eff}}$, PMF, and EDE. However, only EDE also relieves the ${H}_{0}$ tension in the full datasets to below $2\ensuremath{\sigma}$. We discuss how the ${S}_{8}/{\ensuremath{\sigma}}_{8}$ tension is reduced in recent observations. However, even when adopting a strong tension dataset, no model alleviates the ${S}_{8}/{\ensuremath{\sigma}}_{8}$ tension, nor does better than $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ in the combined case of both ${H}_{0}$ and ${S}_{8}/{\ensuremath{\sigma}}_{8}$ tensions.