The present paper considers the model-based and data-driven control of unknown discrete-time linear systems under event-triggering and self-triggering transmission schemes. To this end, we begin by presenting a dynamic event-triggering scheme (ETS) based on periodic sampling, and a discrete-time looped-functional approach, through which a model-based stability condition is derived. Combining the model-based condition with a recent data-based system representation, a data-driven stability criterion in the form of linear matrix inequalities (LMIs) is established, which also offers a way of co-designing the ETS matrix and the controller. To further alleviate the sampling burden of ETS due to its continuous/periodic detection, a self-triggering scheme (STS) is developed. Leveraging precollected input-state data, an algorithm for predicting the next transmission instant is given, while achieving system stability. Finally, numerical simulations showcase the efficacy of ETS and STS in reducing data transmissions as well as practicality of the proposed co-design methods.