Townsend's attached-eddy hypothesis (AEH) provides a theoretical description of turbulence statistics in the logarithmic region in terms of coherent motions that are self-similar with the wall-normal distance (y). This hypothesis was further extended by Perry and coworkers who proposed attached-eddy models that predict the coexistence of the logarithmic law in the mean velocity and streamwise turbulence intensity as well as spectral scaling for the streamwise energy spectra. The AEH can be used to predict the statistical behaviours of wall turbulence, yet revealing such behaviours has remained an elusive task because the proposed description is established within the limits of asymptotically high Reynolds numbers. Here, we show the self-similar behaviour of turbulence motions contained within wall-attached structures of streamwise velocity fluctuations using the direct numerical simulation dataset of turbulent boundary layer, channel, and pipe flows (Re-tau approximate to 1000). The physical sizes of the identified structures are geometrically self-similar in terms of height, and the associated turbulence intensity follows the logarithmic variation in all three flows. Moreover, the corresponding two-dimensional energy spectra are aligned along a linear relationship between the streamwise and spanwise wavelengths (lambda(x) and lambda(z) respectively) in the large-scale range (12y < lambda(x) < 3-4 delta), which is reminiscent of self-similarity. Consequently, one-dimensional spectra obtained by integrating the two-dimensional spectra over the self-similar range show some evidence for self-similar scaling and the possible existence of k(x)(-1) and k(z)(-1) scaling regions in a similar subrange. The present results reveal that the asymptotic behaviours can be obtained by identifying the self-similar coherent structures in canonical wall turbulence, albeit in low-Reynolds-number flows.