  Dynamical dark energy has been recently suggested as a promising and physical
way to solve the 3.4 sigma tension on the value of the Hubble constant $H_0$
between the direct measurement of Riess et al. (2016) (R16, hereafter) and the
indirect constraint from Cosmic Microwave Anisotropies obtained by the Planck
satellite under the assumption of a $\Lambda$CDM model. In this paper, by
parameterizing dark energy evolution using the $w_0$-$w_a$ approach, and
considering a $12$ parameter extended scenario, we find that: a) the tension on
the Hubble constant can indeed be solved with dynamical dark energy, b) a
cosmological constant is ruled out at more than $95 \%$ c.l. by the Planck+R16
dataset, and c) all of the standard quintessence and half of the "downward
going" dark energy model space (characterized by an equation of state that
decreases with time) is also excluded at more than $95 \%$ c.l. These results
are further confirmed when cosmic shear, CMB lensing, or SN~Ia luminosity
distance data are also included. However, tension remains with the BAO dataset.
A cosmological constant and small portion of the freezing quintessence models
are still in agreement with the Planck+R16+BAO dataset at between 68\% and 95\%
c.l. Conversely, for Planck plus a phenomenological $H_0$ prior, both thawing
and freezing quintessence models prefer a Hubble constant of less than 70
km/s/Mpc. The general conclusions hold also when considering models with
non-zero spatial curvature.
