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Cool question.
I think there are at least three: hemisphere, section of an obloid, cone.
Since you did not say face, you appreciate difference between definitions in platonic solids as defined by Euclid and Euler, and the curved solids. A hemisphere has one flat surface and no vertices, but so does a cone. A vertex is defined as the meeting of edges, which are defined as straight in euclidean geometry. Since there are no straight edges coming to a point in a cone (unless you want to talk about infinite edges emanating from the flat surface), there are no vertices on a cone.
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What solid figure has 0 flat surface and 0 vertices?
a sphere has 0 flat serfaces 0 vertices
What solid figure has one flat surface and zero vertices?
cone
What solid figure has 2 flat surfaces 0 vertices?
The solid figure that has 2 flat surfaces and 0 vertices is a cylinder. A cylinder consists of two parallel circular bases (the flat surfaces) connected by a curved surface. The bases do not have any vertices, as they are continuous curves.
What of a solid figure is a flat surface?
A a solid figure on a flat surface is a plane figure (I do believe)
Which solid figure has zero flat surfaces and zero vertices?
A SPHERE has no vertices and no FLAT surfaces
