  Various recent studies proved that cosmological models with a significant
contribution from cold dark matter isocurvature perturbations are still
compatible with most recent data on cosmic microwave background anisotropies
and on the shape of the galaxy power spectrum, provided that one allows for a
very blue spectrum of primordial entropy fluctuations (n_iso > 2). However,
such models predict an excess of matter fluctuations on small scales, typically
below 40 Mpc/h. We show that the proper inclusion of high-resolution high
signal-to-noise Lyman-alpha forest data excludes most of these models. The
upper bound on the isocurvature fraction alpha=f_iso^2/(1+f_iso^2), defined at
the pivot scale k_0=0.05/Mpc, is pushed down to alpha<0.4, while n_iso=1.9+-1.0
(95% confidence limits). We also study the bounds on curvaton models
characterized by maximal correlation between curvature and isocurvature modes,
and a unique spectral tilt for both. We find that f_iso<0.05 (95% c.l.) in that
case. For double inflation models with two massive inflatons coupled only
gravitationally, the mass ratio should obey R < 3 (95% c.l.).
