The wetting kinetics of a drop on a solid surface is measured by observing the movement of the contact line, which is often seen to be unstable, showing a scalloped profile. Many factors have been cited, which, although they can cause instability, can also be eliminated from the experiments, but still the instabilities appear. The basic shape of a spreading drop has a large curvature localized in the vicinity of the contact line as determined by microscopy. It is shown here using linear stability analysis that this curvature can destabilize the contact line region. When the drop profile is disturbed from a basic thickness of h to h + h′, there are two contributions from h′ in the form of added Laplace pressure. One of these is commonly accounted for in the stability analyses. The other is not, and occurs only if the basic shape has a curvature, and the drop has a large curvature near the apparent dynamic contact line, but only for a wetting liquid. This is why instability is not reported in the case of spreading of drops of non-wetting liquids. It also explains why instability gives rise to the changed spreading kinetics of drops that are sometimes reported in the literature, and suggests that as larger curvatures are expected in forced spreading those cases are probably accompanied quite frequently by unstable contact lines.