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Abstract The quadrant analysis of the intense tangential Reynolds stress in plane turbulent channels is generalized to three-dimensional structures (Qs), with special emphasis on the logarithmic and outer layers. Wall-detached Qs are background stress fluctuations. They are small and isotropically oriented, and their contributions to the mean stress cancel. Wall-attached Qs are larger, and carry most of the mean Reynolds stresses. They form a family of roughly self-similar objects that become increasingly complex away from the wall, resembling the vortex clusters in del Álamo et al. ( J. Fluid Mech. , vol. 561, 2006, pp. 329–358). Individual Qs have fractal dimensions of the order of $D= 2$ , slightly fuller than the clusters. They can be described as ‘sponges of flakes’, while vortex clusters are ‘sponges of strings’. The number of attached Qs decays away from the wall, but the fraction of the stress that they carry is independent of their sizes. A substantial fraction of the stress resides in a few large objects extending beyond the centreline, reminiscent of the very large structures of several authors. The predominant logarithmic-layer structure is a side-by-side pair of a sweep (Q4) and an ejection (Q2), with an associated cluster, and shares dimensions and stresses with the conjectured attached eddies of Townsend ( J. Fluid Mech. , vol. 11, 1961, pp. 97–120). Those attached eddies tend to be aligned streamwise from each other, located near the side walls between the low- and high-velocity large-scale streaks, but that organization does not extend far enough to explain the very long structures in the centre of the channel.