  The Galileon model is a tensor-scalar theory of gravity which explains the
late acceleration of the Universe expansion with no instabilities and recovers
General Relativity in the strong field limit. Most constraints obtained so far
on Galileon model parameters from cosmological data were derived for the
limited subset of tracker solutions and reported tensions between the model and
data. This paper presents the first exploration of the general solution of the
Galileon model, which is confronted against recent cosmological data for both
background observables and linear perturbations, using Monte-Carlo Markov
chains. As representative scenarios of the Galileon models, we study the full
Galileon model with disformal coupling to matter and the uncoupled cubic
Galileon model. We find that the general solution of the full Galileon model
provides a good fit to CMB spectra, while the cubic Galileon model does not.
When extending the comparison to BAO and SNIa data, even the general solution
of the full Galileon model fails at providing a good fit to all datasets
simultaneously. Tensions remain if the models are extended with an additional
free parameter, such as the sum of active neutrino masses or the normalization
of the CMB lensing spectrum. Finally, the multi-messenger observation of
GW170817 is also discussed in the framework of the scenarios considered. The
time delay between the gravitational signal and its electromagnetic counterpart
was computed \textit{a posteriori} in every scenario of the full Galileon model
cosmological fit chains and found to be ruled out by this observation.
