The Icelandic mantle plume has had a profound influence on the development of the North Atlantic region over its 64 Myr existence. Long-wavelength free-air gravity anomalies and full waveform tomographic studies suggest that the planform of the plume is highly irregular, with up to five fingers of hot asthenosphere radiating away from Iceland beneath the lithospheric plates. Two of these fingers extend beneath the British Isles and southern Scandinavia, where departures from crustal isostatic equilibrium and anomalous uplift have been identified. In this study, the spatial extent of present-day dynamic support associated with the Icelandic plume is investigated using receiver function analysis. Teleseismic events recorded at nine temporary and 59 permanent broadband, three-component seismometer stations are used to calculate 3864 P-to-S crustal receiver functions. The amplitude and arrival time of particular converted phases are assessed, and H-$k$ stacking is applied to estimate bulk crustal properties. Sub-selections of receiver functions are jointly inverted with Rayleigh wave dispersion data to obtain crustal VS profiles at each station. Both inverse- and guided forward modelling techniques are employed, as well as a Bayesian, trans-dimensional algorithm. Moho depths thus obtained are combined with seismic wide-angle and deep reflection data to produce a comprehensive crustal thickness map of northwestern Europe. Moho depth is found to decrease from southeast (37 km) to northwest (26 km) in the British Isles and from northeast (46 km) to southwest (29 km) in Scandinavia, and does not positively correlate with surface elevation. Using an empirical relationship, crustal shear wave velocity profiles are converted to density profiles. Isostatic balances are then used to estimate residual topography at each station, taking into account these novel constraints on crustal density. Areas of significant residual topography are found in the northwestern British Isles (1400 m), southwestern Scandinavia (464 m) and Denmark (620 m), with convective support from the Icelandic plume as its most likely source. Finally, the irregular planform of the Icelandic plume is proposed to be a manifestation of radial viscous fingering due to a Saffman-Taylor instability. This fluid dynamical phenomenon occurs when less viscous fluid is injected into a layer of more viscous fluid. By comparing the thermal and convective characteristics of the plume with experimental and theoretical results, it is shown that viscous fingering could well explain the present-day distribution of plume material.