In this paper, we present an efficient method for constraining both thermal and nonthermal dark matter (DM) scenarios with the Lyman-α forest based on a simple and flexible parametrization capable of reproducing the small-scale clustering signal of a large set of noncold 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 analysis. We obtain a marginalized upper limit on the largest possible scale at which a power suppression induced by nearly any NCDM scenario can occur, i.e., α<0.03 Mpc/h (2σ 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 parametrization 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 appears to be in mild tension (∼2σ C.L.) with the data compared to nonthermal scenarios. We show that a DM fluid composed by both a warm (thermal) and a cold component is also in tension with the Lyman-α forest, at least for large α values. This is the first study that allows us to probe the linear small-scale shape of the DM power spectrum for a large set of NCDM models.