Assume throughout that $X$ is a normal rational variety over $\mathbb{C}$. Is there a way to determine whether or not $X$ is a toric variety? I am particularly interested in the following examples.
1) $X$ is an affine hypersurface of dimension 3 (e.g. $V(X^2 + Y^2 + Z^3 + T^3) \subset \mathbb{A}^4)$
2) $X$ is an affine hypersurface $(\dim X \geq 2$)
3) $X$ is a projective hypersurface $(\dim X \geq 2$)
4) $X$ is a weighted projective hypersurface in weighted projective space $(\dim X \geq 2$)