  The phenomenological parametrizations of dark-energy (DE) equation of state
can be very helpful, since they allow for the investigation of its cosmological
behavior despite the fact that its underlying theory is unknown. However,
although there has been a large amount of research on DE parametrizations which
involve two or more free parameters, the one-parameter parametrizations seem to
be underestimated. We perform a detailed observational confrontation of five
one-parameter DE models, with observational data from cosmic microwave
background (CMB), Joint light-curve analysis sample from Supernovae Type Ia
observations (JLA), baryon acoustic oscillations (BAO) distance measurements,
and cosmic chronometers (CC). We find that all models favor a phantom DE
equation of state at present time, while they lead to $H_0$ values in perfect
agreement with its direct measurements and therefore they offer an alleviation
to the $H_0$-tension. Finally, performing a Bayesian analysis we show that
although $\Lambda$CDM cosmology is still favored, one-parameter DE models have
similar or better efficiency in fitting the data comparing to two-parameter DE
parametrizations, and thus they deserve a thorough investigation.
