We construct Colding-Minicozzi limit minimal laminations in open domains in 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 L is a locally finite collection S(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 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 H n in closed compact neighborhoods N n of the circle that converge as n → ∞ to R 3 − x 3 -axis. In this case, we prove that the H n converge on compact sets to the foliation of R 3 − x 3 -axis by vertical half planes with boundary the x 3 -axis and with S 1 (1) as the singular set of C 1 -convergence. The H n have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.