Nonlinear Rayleigh-Taylor instability of a heavy, infinitely conducting fluid, supported against gravity by a uniform magnetic field in the vacuum, is studied for three-dimensional disturbances using the method of multiple time-scales. The three-dimensional problem can be reduced to two dimensions as it is found that an instability present for a three-dimensional disturbance of a given wavelength, for a given equilibrium magnetic field, is also present for a two-dimensional disturbance of the same wavelength propagating along an equilibrium magnetic field of lower strength. The instability is studied for both standing and progressive waves. Although in the linear stability problem the instability growth rate for a progressive wave of a given wavelength is equal to that for a stationary wave of the same wavelength, in the nonlinear instability problem studied here these waves are found to have different growth rates. Our results are compared and contrasted with those for the two-dimensional instability problem studied earlier.