Monte Carlo simulations have been performed to examine the equilibrium structure of single chains of random AB copolymers confined between two solid surfaces. Monomers of type A are attracted to the surfaces with a strength epsilon, while monomers of type B are repelled from the surface with the same strength. The copolymer chains are not permanently attached to the surfaces, so they can undergo transitions from adsorbed states, localized near one of the two solid surfaces, to desorbed states, delocalized in the central region between the surfaces. For chains that are strongly adsorbed, the disparity between the surface affinities of A and B segments leads to microphase segregation, with narrow A-rich regions directly adjacent to the surfaces and wider B-rich regions farther away from the surfaces. As a function of the segment-surface interaction energy and the copolymer composition, we calculate monomer density profiles and order parameters which characterize the adsorption-desorption transition and the extent of microphase ordering in the adsorbed states. The simulation results are in excellent qualitative agreement with recent theoretical calculations for confined random copolymers. Our model system may also be considered as a crude representation of polyampholyte chains subject to strongly screened electrostatic interactions with a charged surface. The Monte Carlo simulations indicate, in accord with theoretical predictions and experimental findings, that a polyampholyte can adsorb to a charged surface even when the polymer chain and the surface carry net charges of the same sign.