Probing seismic anisotropy of the lithosphere provides valuable clues on the fabric of rocks. We present a 3-D probabilistic model of shear wave velocity and radial anisotropy of the crust and uppermost mantle of Europe, focusing on the mountain belts of the Alps and Apennines. The model is built from Love and Rayleigh dispersion curves in the period range 5 to 149 s. Data are extracted from seismic ambient noise recorded at 1521 broadband stations, including the AlpArray network. The dispersion curves are first combined in a linearised least squares inversion to obtain 2-D maps of group velocity at each period. Love and Rayleigh maps are then jointly inverted at depth for shear wave velocity and radial anisotropy using a Bayesian Monte-Carlo scheme that accounts for the trade-off between radial anisotropy and horizontal layering. The isotropic part of our model is consistent with previous studies. However, our anisotropy maps differ from previous large scale studies that suggested the presence of significant radial anisotropy everywhere in the European crust and shallow upper mantle. We observe instead that radial anisotropy is mostly localized beneath the Apennines while most of the remaining European crust and shallow upper mantle is isotropic. We attribute this difference to trade-offs between radial anisotropy and thin (hectometric) layering in previous studies based on least-squares inversions and long period data (>30 s). In contrast, our approach involves a massive dataset of short period measurements and a Bayesian inversion that accounts for thin layering. The positive radial anisotropy (VSH > VSV) observed in the lower crust of the Apennines cannot result from thin layering. We rather attribute it to ductile horizontal flow in response to the recent and present-day extension in the region.