This article studies the event-triggered control problem for nonholonomic systems in the chained form with disturbances and drift uncertain nonlinearities. Both the cases of state-feedback and output-feedback are investigated. To address the effects of nonholonomic constraints in event-triggered control, a new systematic design integrating a state-scaling technique and set-valued maps are proposed. A crucial strategy is to transform the event-triggered control system into an interconnection of multiple input-to-state stable systems, to which the cyclic-small-gain theorem is applied for event-based controller synthesis. It is shown that the cyclic-small-gain-based design scheme leads to Zeno-free event-triggered controllers. For the output-feedback case, a new nonlinear observer is designed to deal with the sampling errors. Interestingly, the obtained results are new even if the plant model is disturbance-free. Both numerical and experimental results validate the efficiency of the proposed cyclic-small-gain-based event-triggered control methodology.