In this paper, adaptive neural network (NN) control is investigated for a class of single-input single-output (SISO) discrete-time unknown non-linear systems with general relative degree in the presence of bounded disturbances. Firstly, the systems are transformed into a causal state space description, adaptive state feedback NN control is presented based on Lyapunov's stability theory. Then, by converting the systems into a causal input output representation, adaptive output feedback NN control is given. Finally, adaptive NN observer design and observer-based adaptive control are presented under the assumption of persistent excitation ( PE). All the control schemes avoid the so-called controller singularity problem in adaptive control. By suitably choosing the design parameters, the closed-loop systems are proven to be semi-globally uniformly ultimately bounded (SGUUB). Simulation studies show the effectiveness of the newly proposed schemes.