The double cone is not a surface.

If you remove a point from an open subset in $\mathbb{R}^2$, it remains connected. A surface patch around the vertex has to give a homeomorphism to an open subset of $\mathbb{R}^2$. However, any open neighborhood of the vertex has the property that if you remove the vertex it becomes disconnected. Thus it cannot be homeomorphic to a subset of $\mathbb{R}^2$.

Note that if S where to be a regular surface, then it would be, in an open nhood of $(0,0,0)$ in $S$, the graph of differentiable function of the form: $x=f(y,z)$, $y=g(x,z)$ or $z=h(x,y)$. But that obviously cannot be the case, once the projections of S onto the coordinate planes aren't injective.