This paper investigates the leader-following consensus of general linear singular multi-agent systems. A fully distributed adaptive event-triggered control protocol is first proposed by using the relative state estimate information between neighboring agents. Moreover, the proposed protocol does not require continuous communication between neighbors, which greatly alleviate the negative impact of communication. In addition, a novel adaptive gain is proposed, which will avoid using the Laplace of communication graph. Zeno behavior is excluded by proving that the inter-event times are lower bounded by a positive constant. Finally, a numerical simulation is proposed to verify the effectiveness and reliability of the protocol.