a square
a rectange
a triangle
a hexagon
an octogon
a polygon
If you mean 1 flat surface and 1 vertex, then the only thing I can think of is a cone.
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Cone
A geometric shape that has one flat surface and one vertex is a cone. The flat surface is the circular base, while the vertex is the pointed tip at the opposite end of the cone. This shape is commonly seen in everyday objects like ice cream cones and traffic cones.
The object described is a cone. A cone has one vertex (the point where the flat face meets the curved surface) and one flat face (the circular base). Due to its shape, a cone can roll along a flat surface smoothly.
a cone has one flat surface one vertex and no edge.
The only thing I can think of is a cone.
It has a flat base, a curved surface area, a circular edge and one vertex.
The solid that has one flat face, one curved face, one edge, and one vertex is a cone. The flat face is the circular base, the curved face is the conical surface, the edge is the circular boundary where the base meets the curved surface, and the vertex is the pointed tip of the cone.
A cardioid is one possibility. See link, below.
A shape with 1 vertex, 1 edge, 1 flat face, and 1 curved face is a cone. The vertex is the tip of the cone, the edge is the circular base where the flat face meets the curved surface, and the flat face is the circular base itself. The curved face extends from the base to the vertex.
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.