Rupture of liquid sheets of power-law fluids surrounded by a gas is analysed under the competing influences of pressure due to van der Waals attraction, inertia, viscous stress and capillary pressure due to surface tension. Results of a combined theoretical and computational study are presented over the entire range of parameters governing the thinning of a power-law fluid of power-law exponent $0 < n \le 1$ ($n=1$: Newtonian fluid) and Ohnesorge number $0 \le Oh < \infty$, where $Oh \equiv \mu _0/\sqrt {\rho h_0 \sigma }$, and $\mu _0, \rho, h_0$ and $\sigma$ stand for the zero-deformation-rate viscosity, density, the initial sheet half-thickness and surface tension, respectively. The dynamics in the vicinity of the space–time singularity where the sheet ruptures is asymptotically self-similar, and thus the variation with time remaining until rupture $\tau \equiv t_R - t$, where $t_R$ is the time instant $t$ at which the sheet ruptures, of sheet half-thickness, lateral length scale and lateral velocity is determined analytically and confirmed by simulations. For sheets for which inertia is negligible ($Oh^{-1}=0$), two distinct viscous scaling regimes are found, one for $0.58 < n \le 1$ and the other for $n \le 0.58$. The thinning dynamics of inviscid sheets ($Oh = 0$) is identical to that of Newtonian ones. For real fluids for which neither viscosity nor inertia is negligible, it is shown that the aforementioned creeping and inertial flow regimes are transitory and the thinning of power-law sheets exhibits a remarkably richer set of scaling transitions compared with Newtonian sheets.