In this paper, we investigate the properties of involutes of singular
spherical curves. In general, the involute of a regular spherical curve
has singularities, hence we consider Legendre curves in the unit
spherical bundle. By using the moving frame and the curvature of fronts,
we define involutes of fronts in the Euclidean 2-sphere. We give some
properties of involutes at singular points. Moreover, we consider the
relationships between evolutes and involutes of fronts without
inflection points and give a kind of four vertices theorem. Furthermore,
by the definition of pedal curves, we define contrapedal curves of
fronts in the Euclidean 2-sphere and give some relationships between
them.