Abstract: A broad range of single field models of inflation are analyzed in light of all relevant recent cosmological data, checking whether they can lead to the formation of long lived Primordial Black Holes (PBHs) as candidate for dark matter. To that end we calculate the spectral index of the power spectrum of primordial perturbations as well as its first and second derivatives. PBH formation is possible only if the spectral index increases significantly at small scales. Since current data indicate that the first derivative of the spectral index is negative at the pivot scale, PBH formation is only possible in the presence of a sizable and positive second derivative (running of the running") of the spectral index. Among the studied models, only the running-mass model allows PBH formation.