At high angular frequencies, beyond the damping tail of the primary cosmic microwave background (CMB) power spectrum, the thermal Sunyaev–Zel’dovich (tSZ) effect constitutes the dominant signal in the CMB sky. The tSZ effect is caused by large-scale pressure fluctuations in the baryonic distribution in the universe, such that its statistical properties provide estimates of corresponding properties of the projected 3D pressure fluctuations. The power spectrum of the tSZ is a sensitive probe of the amplitude of density fluctuations, and the bispectrum can be used to separate the bias associated with the pressure. The bispectrum is typically probed with its one-point real-space analogue, the skewness. In addition to the ordinary skewness the morphological properties, as probed by the well-known Minkowski functionals, also require the generalized one-point statistics, which at the lowest order are identical to the generalized skewness parameters. The concept of generalized skewness parameters can be further extended to define a set of three associated generalized skew-spectra. We use these skew-spectra to probe the morphology of the tSZ sky or the y-sky. We show how these power spectra can be recovered from the data in the presence of an arbitrary mask and noise templates using the well known pseudo-Cl approach for arbitrary beam shape. We also employ an approach based on the halo model to compute the tSZ bispectrum. The bispectrum from each of these models is then used to construct the generalized skew-spectra. We consider the performance of an all-sky survey with Planck-type noise and compare the results against a noise-free ideal experiment using a range of smoothing angles. We find that the skew-spectra can be estimated with very high signal-to-noise ratio from future frequency-cleaned tSZ maps that will be available from experiments such as Planck. This will allow their mode-by-mode estimation for a wide range of angular frequencies l and will help us distinguish them from other sources of non-Gaussianity.