Cubic surface parametrized by (s,t,s²+s²t+t³) with parabolic curve
The contour curve is first empty, and becomes a "lip" curve with 2 cusps.
The lip event happens when the contour is an isolated point.

Parabolic surface of the example surface above
The parabolic surface is ruled by the unique principal tangents at the parabolic points of a given surface.

Cubic surface parametrized by (s,t,s²-s²t+t³) with parabolic curve
Two cusps come together and form a tacnode, which then breaks into 2 smooth branches.
The beak-to-beak event happens exactly at the tacnode.

Parabolic surface of the example surface above
The parabolic surface is ruled by the unique principal tangents at the parabolic points of a given surface.

Quartic surface parametrized by (s,t,s²+3st+t⁴) with flecnodal curve and 2 flecnodal lines at a point
The contour is first smooth, and becomes a "swallowtail" curve with 2 cusps and a node.
The swallowtail event happens when the 2 cusps and the node come together to a higher order singularity.

Flecnodal surface of the example surface above
The flecnodal surface is ruled by the flecnodal lines of a given surface.