We construct Colding–Minicozzi limit minimal laminations in open domains in \({\mathbb{R}}^3\) with the singular set of C^{1}-convergence being any properly embedded C^{1,1}-curve. By Meeks’ C^{1,1}-regularity theorem, the singular set of convergence of a Colding–Minicozzi limit minimal lamination \({\mathcal{L}}\) is a locally finite collection \(S({\mathcal{L}})\) of C^{1,1}-curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding–Minicozzi limit minimal lamination. In the case the curve is the unit circle \({\mathbb{S}}^1(1)\) in the (x_{1}, x_{2})-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli H_{n} of finite total curvature which contain the circle. The complete minimal surfaces H_{n} contain embedded compact minimal annuli \(\overline{H}_n\) in closed compact neighborhoods N_{n} of the circle that converge as \(n \to \infty\) to \(\mathbb {R}^3 - x_3\) -axis. In this case, we prove that the \(\overline{H}_n\) converge on compact sets to the foliation of \(\mathbb {R}^3 - x_3\) -axis by vertical half planes with boundary the x_{3}-axis and with \({\mathbb{S}}^1(1)\) as the singular set of C^{1}-convergence. The \(\overline{H}_n\) have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.