= Solution
For every $t\in T_N$, the torus action map
$$
x\longmapsto t\cdot x
$$
is an <automorphism> of $X_\Sigma$. It induces isomorphisms of <local rings>, so it carries smooth points to smooth points and singular points to singular points. The singular locus is consequently invariant under $T_N$. If it contains one point $x$, it contains the entire orbit $T_N\cdot x$; hence the <torus-invariance of the singular locus of a toric variety> proves that $X_\Sigma^{\mathrm{sing}}$ is a union of the orbits in the <orbit-cone correspondence>.
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