Many vision applications require robust point detection as a preliminary task. This can be efficiently done in Gaussian scale-space, with Harris-Laplacian or Lindeberg detectors. Yet, such a uniform smoothing may be a drawback for some applications. Continuous wavelet or curvelet domains have shown to be well adapted to 1D and 2D singularity detection and are therefore an alternative to Gaussian scale-space. Discretization makes the wavelet transform loose its translation and dilation invariance, which is particularly true for critically sampled transforms. In this paper, we investigate discrete wavelet transforms for points detection, show that a redundant transform such as contourlet transform yield to more robust points than a critically sampled one, and compare results with Harris-Laplacian and Lindeberg point detectors.