I was considering the following question for 3D curves: Does zero curvature imply zero torsion?
I think it's reasonable, because zero curvature implies the curve is a straight line, which lies in a plane, making the torsion zero.
However, as I checked the definitions, the Frenet-Serret frame is not even defined if $\kappa=0$ (even a single point with zero curvature seems problematic). What is the procedure if the curvature vanishes at a point? Does it mean trouble? What happens to the FS frame?