Technically, there's no shape with no surface and no vertices, but a sphere is close. It has no vertices but has just one surface.
A cone.
If you are a cube, one of the sides is called a face. Same with a cone, the flat part is a face.
A cone would fit the given description
An ovoid (egg-shape) is one possible answer. A smooth blob (to use a very technical term!) is another.
none!all shapes are flat
Hemisphere
Technically, there's no shape with no surface and no vertices, but a sphere is close. It has no vertices but has just one surface.
cone
a cone has one flat surface one vertex and no edge.
A tetrahedron
a cone.
It has a flat base, a curved surface area, a circular edge and one vertex.
Circle
There can be no such shape. If there were a shape with n flat surfaces, then if you sliced off one vertex, you would have a shape with n+1 flat surfaces. So it is always possible to add one more flat surface.
A cone.
A Cylinder