  The main question studied in this paper concerns the weak-coupling behavior
of the geometrically induced bound states of singular Schr\"odinger operators
with an attractive $\delta$ interaction supported by a planar, asymptotically
straight curve $\Gamma$. We demonstrate that if $\Gamma$ is only slightly bent
or weakly deformed, then there is a single eigenvalue and the gap between it
and the continuum threshold is in the leading order proportional to the fourth
power of the bending angle, or the deformation parameter. For comparison, we
analyze the behavior of a general geometrical induced eigenvalue in the
situation when one of the curve asymptotes is wiggled.
