  Recent measurements of the Cosmic Microwave Background (CMB) by the Planck
Collaboration have produced arguably the most powerful observational evidence
in support of the standard model of cosmology, i.e. the spatially flat
$\Lambda$CDM paradigm. In this work, we perform model selection tests to
examine whether the base CMB temperature and large scale polarization
anisotropy data from Planck 2015 (P15) prefer any of eight commonly used
one-parameter model extensions with respect to flat $\Lambda$CDM. We find a
clear preference for models with free curvature, $\Omega_\mathrm{K}$, or free
amplitude of the CMB lensing potential, $A_\mathrm{L}$. We also further develop
statistical tools to measure tension between datasets. We use a Gaussianization
scheme to compute tensions directly from the posterior samples using an
entropy-based method, the surprise, as well as a calibrated evidence ratio
presented here for the first time. We then proceed to investigate the
consistency between the base P15~CMB data and six other CMB and distance
datasets. In flat $\Lambda$CDM we find a $4.8\sigma$ tension between the base
P15~CMB data and a distance ladder measurement, whereas the former are
consistent with the other datasets. In the curved $\Lambda$CDM model we find
significant tensions in most of the cases, arising from the well-known low
power of the low-$\ell$ multipoles of the CMB data. In the flat $\Lambda$CDM
$+A_\mathrm{L}$ model, however, all datasets are consistent with the base
P15~CMB observations except for the CMB lensing measurement, which remains in
significant tension. This tension is driven by the increased power of the CMB
lensing potential derived from the base P15~CMB constraints in both models,
pointing at either potentially unresolved systematic effects or the need for
new physics beyond the standard flat $\Lambda$CDM model.
