Scattering transforms have been successfully used to describe dust polarisation for flat-sky images. This paper expands this framework to noisy observations on the sphere with the aim of obtaining denoised Stokes Q and U all-sky maps at 353 GHz, as well as a non-Gaussian model of dust polarisation, from the Planck data. To achieve this goal, we extended the computation of scattering coefficients to the HEALPix pixelation and introduced cross-statistics that allowed us to make use of half-mission maps as well as the correlation between dust temperature and polarisation. Introducing a general framework, we developed an algorithm that uses the scattering statistics to separate dust polarisation from data noise. The separation was validated on mock data before it was applied to the SRoll2Planck maps at Nside = 256. The validation shows that the statistics of the dust emission, including its non-Gaussian properties, are recovered until ℓmax ~ 700, where, at high Galactic latitudes, the dust power is weaker than that of the dust by two orders of magnitude. On scales where the dust power is weaker than one-tenth of the power of the noise, structures in the output maps have comparable statistics, but are not spatially coincident with those of the input maps. Our results on Planck data are significant milestones opening new perspectives for statistical studies of dust polarisation and for the simulation of Galactic polarised foregrounds. The Planck denoised maps are available (see http://sroll20.ias.u-psud.fr/sroll40_353_data.html) together with results from our validation on mock data, which may be used to quantify uncertainties.