  We construct Colding-Minicozzi limit minimal laminations in open domains in
$\rth$ 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 ${\cal L}$ is a
locally finite collection $S({\cal L})$ 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 $\esf^1(1)$
in the $(x_1, x_2)$-plane, the classical Bj\"orling 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 $\bar{H}_n$ in closed compact neighborhoods $N_n$ of the
circle that converge as $n \to \infty$ to $\rth - x_3$-axis. In this case, we
prove that the $\bar{H}_n$ converge on compact sets to the foliation of $\rth -
x_3$-axis by vertical half planes with boundary the $x_3$-axis and with
$\esf^1(1)$ as the singular set of $C^1$-convergence. The $\bar{H}_n$ have the
appearance of highly spinning helicoids with the circle as their axis and are
named {\em bent helicoids}.
