Atomic-resolution structural and spectroscopic characterization techniques (scanning transmission electron microscopy and electron energy loss spectroscopy) are combined with nanoscale electrical measurements (conductive atomic force microscopy) to study at the atomic scale the properties of graphene grown epitaxially through the controlled graphitisation of Si-face and C-face hexagonal SiC(0001) substrates by high temperature annealing. A scanning transmission electron microscopy analysis, carried out at 60KeV of beam energy, below the knock-on threshold for carbon to ensure no damage is imparted to the film by the electron beam, demonstrates that the buffer layer present on the planar SiC(0001) Si-face delaminates from it on the (11-2n) facets of SiC surface steps, In addition, electron energy loss spectroscopy reveals that the delaminated layer has a similar electronic configuration to purely sp2-hybridized graphene. A thin amorphous film is found on the C-face, instead, which strongly suppresses epitaxy with the SiC substrate. Structurally, the amorphous area is inhomgeneous, as its Si-concentration gradually decreases while approaching the first graphene layer, which is purely sp2-hybridized. Based on these features, we discuss differences and similarities between the C-only buffer layer that forms on the Si-face of SiC with respect to the thicker C/Si amorphous film of the C-face.