Fast ice flow is associated with the deformation of subglacial sediment. Seismic shear velocities, Vs, increase with the rigidity of material and hence can be used to distinguish soft sediment from hard bedrock substrates. Depth profiles of Vs can be obtained from inversions of Rayleigh wave dispersion curves, from passive or active-sources, but these can be highly ambiguous and lack depth sensitivity. Our novel Bayesian transdimensional algorithm, MuLTI, circumvents these issues by adding independent depth constraints to the inversion, also allowing comprehensive uncertainty analysis. We apply MuLTI to the inversion of a Rayleigh wave dataset, acquired using active-source (Multichannel Analysis of Surface Waves) techniques, to characterise sediment distribution beneath the frontal margin of Midtdalsbreen, an outlet of Norway's Hardangerjokulen ice cap. Ice thickness (0-20 m) is constrained using co-located GPR data. Outputs from MuLTI suggest that partly-frozen sediment (Vs 500-1000 m s-1), overlying bedrock (Vs 2000-2500 m s-1), is present in patches with a thickness of ~4 m, although this approaches the resolvable limit of our Rayleigh wave frequencies (14-100 Hz). Uncertainties immediately beneath the glacier bed are <280 m s-1, implying that MuLTI cannot only distinguish bedrock and sediment substrates but does so with an accuracy sufficient for resolving variations in sediment properties.