The variability of turbulent momentum flux in neutral and unstable atmospheric boundary layers is characterized by analyzing surface-layer measurements and data from large-eddy simulations (LES). The method involves multiresolution (MR) decomposition of vertical wind and advected variables into eddy fluctuations on different scales. It provides a measure of the amount of flux variability that stems from same-scale correlations and from combinations of different-scale eddy fluctuations. Combining two analysis methods enabled MR component cospectra to be introduced, in order to study the contribution of downward and upward flux on different scales. These component cospectra were used to investigate at which scales most of the upward and downward momentum flux occurs. By using MR spectra, cospectra and flow visualization, this investigation provides insights into turbulence structure and fluxes in neutral and unstable stratification. It is shown that most of the flux variability in the lower part of the boundary layer can be characterized as a combination of larger scale streamwise elongated horizontal wind streaks and smaller scale vertical wind fluctuations. These streaks are found to account for a large part of downward momentum flux at relatively large, energy-containing scales. Most of the upward momentum flux is found to occur at smaller scales. This can be interpreted as showing that upward momentum flux in these conditions is caused by the generation of smaller scale secondary motions when larger scale turbulence elements break down and dissipate. Differences in the height dependence of turbulence structure and momentum flux for neutral and unstably stratified conditions are also investigated and related to the existence of wind streaks and horizontal rolls in these different conditions.