  For singular corank 1 surfaces in $\mathbb R^3$ we introduce a distinguished
normal vector called the axial vector. Using this vector and the curvature
parabola we define a new type of curvature called the axial curvature, which
generalizes the singular curvature for frontal type singularities. We then
study contact properties of the surface with respect to the plane orthogonal to
the axial vector and show how they are related to the axial curvature. Finally,
for certain fold type singularities, we relate the axial curvature with the
Gaussian curvature of an appropriate blow up.
