A pedal curve (a contrapedal curve) of a regular plane curve is the locus of the feet of the perpendiculars from a point to the tangents (normals) to the curve. These curves can be parametrized by using the Frenet frame of the curve. Yet provided that the curve has some singular points, the Frenet frame at these singular points is not well-defined. Thus, we cannot use the Frenet frame to examine pedal or contrapedal curves. In this paper, pedal and contrapedal curves of plane curves, which have singular points, are investigated. By using the Legendrian Frenet frame along a front, the pedal and contrapedal curves of a front are introduced and properties of these curves are given. Then, the condition for a pedal (and a contrapedal) curve of a front to be a frontal is obtained. Furthermore, by considering the definitions of the evolute, the involute, and the offset of a front, some relationships are given. Finally, some illustrated examples are presented.