We combine new Parkes telescope observations of neutral hydrogen (Hi) in the Small Magellanic Cloud (SMC) with an Australia Telescope Compact Array (ATCA) aperture synthesis mosaic to obtain a set of images sensitive to all angular (spatial) scales between 98 arcsec (30 pc) and 4 deg (4 kpc). The new data are used to study the HI spatial power spectrum over a range of contiguous scale sizes wider than those previously achieved in any other galaxy, including our own. The spatial power spectrum closely obeys the relation P(k) ~ k^gamma, with gamma =-3.04 +/- 0.02, similar to values obtained by other authors for our own Galaxy which are in the range gamma =-3.0 to -2.8. This is surprising given the very different morphology, gas-richness, star-formation rate and evolution of the two systems, and may imply similar mechanisms for structure formation. One interpretation of the P(k) power-law is that the interstellar medium (ISM) of the SMC is fractal in nature, consisting of a hierarchy of HI cloud structures created, for example, by homogeneous turbulence. The projected fractal dimension of D_p=1.5 is similar to values obtained by other authors for molecular clouds in the Galaxy in the size range ~ 0.05 to 100 pc. Such a model is consistent with a low space-filling factor for the neutral gas. A kinematic study of the HI data reveals the existence of three supergiant shells which were previously undetectable in the ATCA data alone. These shells have diameters up to 1.8 kpc and require energies (in the standard supernova-driven models) up to 2x10^54 erg. The structure and evolution of the ISM in the SMC are heavily influenced by the formation of these supergiant shells.