In 1872, Gaston Darboux defined a family of curves on surfaces in the 3-dimensional Euclidean space $E^3$ which are preserved by the action of the Möbius group and share many properties with geodesics. Here, we study the Darboux curves from a dynamical viewpoint on special canal surfaces, quadrics and some Darboux cyclides. We also describe the generic behavior of Darboux curves near ridge points (zig-zag and beak-to-beak).