Singular cylinder over a nodal curve
= Singular cylinder over a nodal curve
The affine surface
$$
V\bigl(y^2-x(x-1)^2\bigr)\subseteq\mathbb A^3_{x,y,z}
$$
is birational to $\mathbb A^2$ and has singular locus $\{(1,0,z):z\in k\}$, an irreducible affine line.