I think the answer is yes by Proposition 2.2 of Do Carmo, which states the pre-image of regular values of functions of the form $f:\mathbb{R}^3\rightarrow \mathbb{R}$ is a regular surface.
$$S=\{(x,y,z)\in\mathbb{R}^3:x^3+y^3-z^3=1 \}$$
Consider $F(x,y,z)=x^3+y^3-z^3-1$. We have $0$ is a regular value of $F$ since the partial derivatives of $F$ vanish simultaneously only at (0,0,0). Since $F^{-1}(0)=S$, we conclude $S$ is a regular surface.
Is this a valid proof? I’m pretty sure it is, but my notes say this is not a regular surface; I think it is a typo.
Yes it is a regular surface, I recently did a similar question (just difference in the powers of $x,y$ and $z$) in one of my labs. I am posting a pic from the notes: