The classical limit self-consistent field theory of laterally homogeneous polymer brushes is generalized to laterally inhomogeneous melt systems, without imposing the Alexander-de Gennes approximation. This enables the appropriate description of the chain end distribution and confirmations, based on which the free energy expansion is obtained. The brushes are found to be always stable against height fluctuations. For grafted polymers, the equilibrium height profile and the height fluctuation spectrum strongly depend on the lateral wave vector q and the average surface coverage sigma. The structure factor (especially at large q) is qualitatively different from existing predictions by Fredrickson et al. and by Solis and Pickett based on the Alexander-de Gennes approximation, the latter of which even leads to lateral instability at small surface tension and/or large surface coverage, which is clearly an artifact of the approximation. For tethered polymers with lateral mobility, the height fluctuations exhibit a peak at q = 0 and a correlation length alpha N(1/2-)sigma-(1/2) (N being the polymerization index), whereas the surface coverage fluctuations are dominant over length scales similar to N(1/2)l (l being the statistical segment length). The fluctuations of the two quantities are correlated at small q, but are decoupled at large q.