This study focused on the asynchronous event-triggered output-feedback controller design problem for discrete-time singular Markov jump systems (MJSs). A hidden Markov model (HMM) was employed to estimate the system mode, which cannot always be ideally detected in practice. Because the full state is also difficult to obtain in practical scenarios, an output-feedback control scheme was used. In addition, an HMM-based event-triggered mechanism was also employed in the design of the controller to reduce the communication burden of the networked system. Sufficient conditions for the stochastic admissibility of a closed-loop singular MJS with a prescribed H∞ performance index were established using the Lyapunov functional technique. Finally, design procedures for an asynchronous event-triggered controller were summarized as a linear-matrix-inequality-based optimization algorithm. Two examples were considered to verify the effectiveness of the asynchronous event-triggered output-feedback controller design method.