In this paper, six new event-triggered neuro-adaptive control (ETNAC) schemes are presented for uncertain linear systems. Novelty of this paper lies in (i) the construction of the proposed ETNAC schemes, (ii) the design of event-triggering conditions, and (iii) the design of an observer called the modified state observer (MSO). In the proposed schemes, the MSO, the controller, and the event-triggering mechanisms are constructed and organized in a way such that they provide the control system designer with flexibility to choose between the one-way or two-way data exchange and also between the dynamic or static triggering conditions. The event-triggering conditions are designed on the basis of real performance parameters, such as the estimation/tracking errors that render control updates more on actual system events instead of the often-used extended time sampling. Another unique feature of ETNAC is its online uncertainty approximation capability even during inter-event times, which makes the controller robust and efficient. This part is developed with the help of an artificial neural network (ANN) and a polynomial regression-based MSO. The MSO formulations have two tunable gains, which allow fast uncertainty estimation without inducing high frequency oscillations, even while the system is in a transient state. Lyapunov analysis is used to show the stability of the system as well as to develop the event-triggering conditions. Effectiveness of the proposed controllers is demonstrated using benchmark numerical examples.