Considerable amount of work has been accomplished on the fine structure of the near wall turbulence since the beginning of 1960’s with the experimental finding of coherent structures and the process labeled at this time as ‘bursting’. The direct numerical simulations have subsequently contributed to a better understanding at the end of 1980’s through the interrelationships between the coherency and the vortical structures difficult to determine experimentally. More or less sophisticated detection and identification schemes have been proposed so far. However, the question remains open and some new representations of the fluctuating signals may be helpful both for practical and fundamental reasons. We apply the ‘instantaneous amplitude-frequency’ (phase) formulation to the near wall turbulence here. This representation has been widely used so far, both in particle physics and signal analysis, but never been applied to turbulence to the author’s knowledge. We combine it with a particular class of ‘wavelet transforms used for edge detection’ in image analysis. That allows us to objectively define the rapid variations in time or space of the velocity fluctuations, often associated with the bursting process. These strong local time-space gradients are referred as singularities. One of the main aims of this research is to study the locally oscillating zones of the streamwise singularities in the buffer layer. These regions correspond to the meandering of the streaks, a phenomenon, which is fundamental in the understanding of the regeneration process of the quasi-streamwise vortices. The instantaneous frequency-amplitude representation is quite efficient to achieve this goal. We are able to clearly show that the streaks correspond to the localized zones that have non-linear oscillators behaviors with: (i) significant constant phase zones at each scale for the streamwise and only streamwise gradients of the streamwise velocity components; (ii) smoothly oscillating singularities; (iii) large amplitudes; (iv) weak intermittency. We also note that (i) meandering of the streaks is present at small scales, (ii) the small-scale intermittence increases significantly when they are suppressed, (iii) there are identifiable regular oscillating zones in streamwise gradients of the instantaneous longitudinal velocity at small scales, and (iv) the spanwise singularities are significantly less intermittent. The study will be extended to the whole wall layer for all velocity components. Multi-fractal analysis will be applied to the experimental data in the constant phase zones. The Lipschitz measure of regularity of the singularities will be determined and presented. [abstract only., no pdf]