Planar curve singularities relative to a smooth boundary

We study curve singularities in a smooth surface relative to a smooth boundary curve. We consider the semiuniversal deformations and equisingular deformations of curves with a fixed local intersection number $w$ with the boundary, and prove results on the inclusion relations between ideals describing different deformations. A classification is given of singularities expected to appear in codimension $\leq 3$ in a general family with $w\geq 8$.