Local stable reduction of plane curve singularities

Consider a family of curves over the disc, with smooth fibers except for the central fiber over the origin. By the local stable reduction theorem, after suitable blow-ups and base changes we obtain a family such that the central fiber has reduced normal crossings. This stable central fiber has two parts: the proper transform of the original central fiber and the 'tail'. Which tails arise when the original central fiber is a given plane curve singularity? We address this question using the technique of stable reduction for log surfaces. For certain singularities, we find that weighted plane curves naturally arise as tails.