For \(\displaystyle{S_3}\) : We define \(\displaystyle{f:\mathbb{R}^3\longrightarrow \mathbb{R}}\) by \(\displaystyle{f(x,y,z)=-x^2-y^2+z^2-1}\) .
The function \(\displaystyle{f}\) is smooth on \(\displaystyle{\mathbb{R}^3}\) and we observe that \(\displaystyle{S_{3}=f^{-1}\left(\left\{0\right\}\right)}\) .
Also, \(\displaystyle{f(0,0,1)=0\iff 0\in f(\mathbb{R}^3)}\) . For each \(\displaystyle{\left(x,y,z\right)\in\mathbb{R}^3}\) holds :
\(\displaystyle{\dfrac{\partial{f}}{\partial{x}}(x,y,z)=-2\,x\,\,,\dfrac{\partial{f}}{\partial{y}}(x,y,z)=-2\,y\,\,,\dfrac{\partial{f}}{\partial{z}}(x,y,z)=2\,z}\) with
\(\displaystyle{\dfrac{\partial{f}}{\partial{x}}(x,y,z)=\dfrac{\partial{f}}{\partial{y}}(x,y,z)=\dfrac{\partial{f}}{\partial{z}}(x,y,z)=0\iff \left(x,y,z\right)=\left(0,0,0\right)}\)
but \(\displaystyle{\left(0,0,0\right)\notin f^{-1}\left(\left\{0\right\}\right)}\) since \(\displaystyle{f(0,0,0)=-1\neq 0}\) . So, the number \(\displaystyle{0}\) is
a regular value of \(\displaystyle{f}\) and \(\displaystyle{S_{3}=f^{-1}\left(\left\{0\right\}\right)}\) is a regular surface of \(\displaystyle{\mathbb{R}^3}\) .
For \(\displaystyle{S_{2}}\) : Suppose that \(\displaystyle{S_{2}}\) is a regular surface of \(\displaystyle{\mathbb{R}^3}\) . Then, in a neighborhood of
\(\displaystyle{\left(0,0,0\right)\in S_{2}}\), this surface is a graph-surface and takes one of the following forms :
\(\displaystyle{z=f(x,y)\,,x=g(y,z)\,,y=h(x,z)}\). If \(\displaystyle{x=g(y,z)}\) or \(\displaystyle{y=h(x,z)}\), then restricted to the \(\displaystyle{yz\,,xz}\) planes,
we have \(\displaystyle{z=\pm \left|y\right|}\) or \(\displaystyle{z=\pm \left|x\right|}\), and these functions are not one to one. If \(\displaystyle{z=f(x,y)}\), then
\(\displaystyle{z=\pm \sqrt{x^2+y^2}\,,x\,,y\in\mathbb{R}}\) and this function is not differentiable at \(\displaystyle{\left(x,y\right)=\left(0,0\right)}\).
Therefore, \(\displaystyle{S_{2}}\) is not a regular surface.
For the set \(\displaystyle{S_{1}}\) i am not so sure. I am waiting for your opinion.
I would like to ask a question in the same spirit to this one you proposed here :
Prove that \(\displaystyle{S_{3}}\) is not a connected regular surface.