The decomposition of a signal on the sphere with the steerable wavelet constructed from the second Gaussian derivative gives access to the orientation, signed-intensity and elongation of the signal’s local features. In the present work, the non-Gaussianity of the Wilkinson Microwave Anisotropy Probe (WMAP) temperature data of the cosmic microwave background (CMB) is analysed in terms of the ﬁrst four moments of the statistically isotropic random ﬁelds associated with these local morphological measures, at wavelet scales corresponding to angular sizes between 27.5 arcmin and 30° on the celestial sphere. While no detection is made either in the orientation analysis or in the elongation analysis, a strong detection is made in the excess kurtosis of the signed-intensity of the WMAP data. The non-Gaussianity is observed with a signiﬁcance level below 0.5 per cent at a wavelet scale corresponding to an angular size around 10°, and conﬁrmed at neighbouring scales. This supports a previous detection of an excess of kurtosis in the wavelet coefﬁcient of the WMAP data with the axisymmetric Mexican hat wavelet. Instrumental noise and foreground emissions are not likely to be at the origin of the excess of kurtosis. Large-scale modulations of the CMB related to some unknown systematics are rejected as possible origins of the detection. The observed non-Gaussianity may therefore probably be imputed to the CMB itself, thereby questioning the basic inﬂationary scenario upon which the present concordance cosmological model relies. Taking the CMB temperature angular power spectrum of the concordance cosmological model at face value, further analysis also suggests that this non-Gaussianity is not conﬁned to the directions on the celestial sphere with an anomalous signed-intensity.