The paper reports some wetting singularities related to surfactant adsorption at solid-liquid 1-liquid 2 interfaces which are specifically driven by the time evolution of the concentration C(t) around the triple line as the surfactant diffuses freely from a localized source. Two model systems have been considered: a hydrophobic methyl-terminated surface and a hydrophilic glass plate, both in contact with a squalane drop and surrounded by external bulk water. On the hydrophobic surface: an intermediate plateau was observed over a certain concentration range in the kinetics of the drop parameters [contact angle theta(t) and radius R(t)]. This singularity is shown to arise from the overlap of the linens domains in the individual kinetics gamma[C(t)] at the different interfaces. On the hydrophilic plate, an earlier spreading at low concentration is observed before the drop starts to recede as the local concentration around the triple line increases. This inversion in triple line motion, driven by the bilayer adsorption transition at the glass-water interface, combines with the buoyancy in the late stage. leading to the neck rupture of the confined drop.