= Solution
Take the <weighted projective plane> $X=\mathbb P(1,1,2)$. Its <fan in toric geometry>[fan] in $N=\mathbb Z^2$ has rays generated by
$$
v_0=(-1,-2),\qquad v_1=(1,0),\qquad v_2=(0,1),
$$
and all three two-dimensional cones spanned by adjacent rays. Their union is $N_{\mathbb R}$, so the fan is <complete fan>[complete] and the <toric variety> $X$ is <proper toric variety>[proper].
The determinants of the cones $\langle v_1,v_2\rangle$ and $\langle v_2,v_0\rangle$ have absolute value one, while
$$
|\det(v_0,v_1)|=2.
$$
The <smoothness criterion for a toric variety> therefore shows that the affine chart for $\langle v_0,v_1\rangle$ is singular; it is the cyclic quotient singularity of type $\frac12(1,1)$. Thus $X$ is a singular proper toric surface.
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