An accurate characterization of the inaccessible subglacial environment is key to accurately modelling the dynamic behaviour of ice sheets and glaciers, crucial for predicting sea-level rise. The composition and water content of subglacial material can be inferred from measurements of shear wave velocity (Vs) and bulk electrical resistivity (R), themselves derived from Rayleigh wave dispersion curves and transient electromagnetic (TEM) soundings. Conventional Rayleigh wave and TEM inversions can suffer from poor resolution and non-uniqueness. In this thesis, I present a novel constrained inversion methodology which applies a Markov chain Monte Carlo implementation of Bayesian inversion to produce probability distributions of geophysical parameters. MuLTI (Multimodal Layered Transdimensional Inversion) is used to derive Vs from Rayleigh wave dispersion curves, and its TEM variant, MuLTI-TEM, for evaluating bulk electrical resistivity. The methodologies can include independent depth constraints, drawn from external data sources (e.g., boreholes or other geophysical data), which significantly improves the resolution compared to conventional unconstrained inversions. Compared to such inversions, synthetic studies suggested that MuLTI reduces the error between the true and best-fit models by a factor of 10, and reduces the vertically averaged spread of the Vs distribution twofold, based on the 95% credible intervals. MuLTI and MuLTI-TEM were applied to derive Vs and R profiles from seismic and TEM electromagnetic data acquired on the terminus of the Norwegian glacier Midtdalsbreen. Three subglacial material classifications were determined: sediment (Vs < 1600 m/s, 50 Ωm < R < 500 Ωm), permafrost (Vs > 1600 m/s, R > 500 Ωm) and weathered/fractured bedrock containing saline water (Vs > 1900 m/s, R < 50 Ωm). These algorithms offer a step-change in our ability to resolve and quantify the uncertainties in subsurface inversions, and show promise for constraining the properties of subglacial aquifers beneath Antarctic ice masses. MuLTI and MuLTITEM have both been made publicly available via GitHub to motivate users, in the cryosphere and other environmental settings, for continued advancement.