We present spatially resolved maps of six individually-detected Lyman α haloes (LAHs) as well as a first statistical analysis of the Lyman α (Lyα) spectral signature in the circum-galactic medium of high-redshift star-forming galaxies (−17.5 > M UV > −21.5) using the Multi-Unit Spectroscopic Explorer. Our resolved spectroscopic analysis of the LAHs reveals significant intrahalo variations of the Lyα line profile. Using a three-dimensional two-component model for the Lyα emission, we measured the full width at half maximum (FWHM), the peak velocity shift, and the asymmetry of the Lyα line in the core and in the halo of 19 galaxies. We find that the Lyα line shape is statistically different in the halo compared to the core (in terms of width, peak wavelength, and asymmetry) for ≈40% of our galaxies. Similarly to object-by-object based studies and a recent resolved study using lensing, we find a correlation between the peak velocity shift and the width of the Lyα line both at the interstellar and circum-galactic scales. This trend has been predicted by radiative transfer simulations of galactic winds as a result of resonant scattering in outflows. While there is a lack of correlation between the spectral properties and the spatial scale lengths of our LAHs, we find a correlation between the width of the line in the LAH and the halo flux fraction. Interestingly, UV bright galaxies (M UV < −20) show broader, more redshifted, and less asymmetric Lyα lines in their haloes. The most significant correlation found is for the FWHM of the line and the UV continuum slope of the galaxy, suggesting that the redder galaxies have broader Lyα lines. The generally broad and red line shapes found in the halo component suggest that the Lyα haloes are powered either by scattering processes through an outflowing medium, fluorescent emission from outflowing cold clumps of gas, or a mix of both. Considering the large diversity of the Lyα line profiles observed in our sample and the lack of strong correlation, the interpretation of our results is still broadly open and underlines the need for realistic spatially resolved models of the LAHs.