This is a joint work of Saji, Umehara and Yamada. We talk on the recognition problem of Ak-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role to generalize the authors previous work the geometry of fronts for surfaces. The crucial point to prove our criteria for Ak-singularities is to introduce a suitable parametrization of the singularities called the k-th KRSUY-coordinates. Using them, we can directly construct a versal unfolding for a given singularity. As an application, we prove that a given nondegenerate singular point p on a hypersurface (as a wave front) in $R^{n+1}$ is differentiable right-left equivalent to the $A_{k+1}$-type singular point if and only if the linear projection of the singular set around p into a generic hyperplane $R^n$ is right-left equivalent to the $A_k$-type singular point in $R^n$. Moreover, we shall give a relationship between the normal curvature map and the zig-zag numbers (the Maslov indices) of wave fronts.