This paper addresses the problem of pulmonary microvascular pressure estimation. The "why" and "how" of multiscale approaches in the context of singularity detection are discussed. From a linear viewpoint, the scalogram technique and the multiresolution representation for singularity detection are discussed in general under the wavelet framework. A technique as well as a criterion for the segmentation and the extraction of the region of interest or the optimal scale is proposed. A new nonlinear multiscale technique for singularity detection is also proposed. This multiscale representation is based on rank order filters and on mathematical morphology operators in particular. A scaling parameter is introduced for such operators as in the linear continuous wavelet transform leading to what we call morpholograms. Experiments performed on pulmonary artery pressure transients illustrate how the proposed technique can give a synopsis of the signal characteristics through the scales.