We consider the problem of parameter estimation in the case of observation of the trajectory of diffusion process. We suppose that the drift coefficient has a singularity of cusp-type and the unknown parameter corresponds to the position of the point of the cusp. The asymptotic properties of the maximum likelihood estimator and Bayesian estimators are described in the asymptotics of {\it small noise}, i.e., as the diffusion coefficient tends to zero. The consistency, limit distributions and the convergence of moments of these estimators are established.