  In this paper we present an efficient method for constraining both thermal
and non-thermal Dark Matter (DM) scenarios with the Lyman-$\alpha$ forest,
based on a simple and flexible parametrisation capable to reproduce the small
scale clustering signal of a large set of non-cold DM (nCDM) models. We extract
new limits on the fundamental DM properties, through an extensive analysis of
the high resolution, high redshift data obtained by the MIKE/HIRES
spectrographs. By using a large suite of hydrodynamical simulations, we
determine constraints on both astrophysical, cosmological, and nCDM parameters
by performing a full Monte Carlo Markov Chain (MCMC) analysis. We obtain a
marginalised upper limit on the largest possible scale at which a power
suppression induced by nearly any nCDM scenario can occur, i.e.
$\alpha<0.03~{\rm{Mpc}}/h$ (2$\sigma$ C.L.). We explicitly describe how to test
several of the most viable nCDM scenarios without the need to run any specific
numerical simulations, due to the novel parametrisation proposed, and due to a
new scheme that interpolates between the cosmological models explored. The
shape of the linear matter power spectrum for standard thermal warm DM models
appear to be in mild tension ($\sim 2\sigma$ C.L.) with the data, compared to
non-thermal scenarios. We show that a DM fluid composed by both a warm
(thermal) and a cold component is also in tension with the Lyman-$\alpha$
forest, at least for large $\alpha$ values. This is the first study that allows
to probe the linear small scale shape of the DM power spectrum for a large set
of nCDM models.
