We consider the d-dimensional massless free field localized by a δ-pinning of strength ɛ. We study the asymptotics of the variance of the field (when d= 2), and of the decay-rate of its 2-point function (when d≥ 2), as ɛ goes to zero, for general Gaussian interactions. Physically speaking, we thus rigorously obtain the critical behavior of the transverse and longitudinal correlation lengths of the corresponding d+ 1-dimensional effective interface model in a non-mean-field regime. We also describe the set of pinned sites at small ɛ, for a broad class of d-dimensional massless models.