= Solution
For the affine polynomial $f(x,y)=y^2+y-x^3+x$, a <singular point of an algebraic variety> must satisfy $f=f_x=f_y=0$. Modulo $37$, the derivative equations are $2y+1=0$ and $3x^2=1$. Thus $y=18$ and $x^2=25$, giving candidates $x=5$ and $x=32$. At $(5,18)$, the two sides of the curve equation are both $9$ modulo $37$. At $(32,18)$, the right side is $28$, so that candidate is not on the curve. The point at infinity is nonsingular: for the homogeneous equation $Y^2Z+YZ^2-X^3+XZ^2=0$, the derivative with respect to $Z$ is one at $(0:1:0)$. Therefore the unique <singular point of an algebraic variety> is
$$
\boxed{(5,18)\in\widetilde E_{37}(\mathbb F_{37}).}
$$
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