The information interaction between agents in this paper is described using Markov switching topology, and an adaptive neural-network sliding mode (ANNSM) controller is proposed to address the admissible consensus problem for discrete-time nonlinear singular multi-agent systems (SMASs) with external disturbances. For the reason of approximating the nonlinear part online and overcoming the bounded difficulty, the radial basis function (RBF) based on neural network (NN) is introduced. When designing the state observer, a disturbance observer assisted by an adaptive update law is established for the purpose of estimating the external disturbance and the NN residual error. Next, an ANNSM control method related to the Markov switching topology is proposed based on the designed disturbance observer, and the Lyapunov stability theory demonstrates that the system achieves admissible bounded consensus. Finally, simulation studies on a numerical example and a distributed microgrid model to demonstrate the effectiveness of the proposed controller.