Spike-timing-dependent plasticity (STDP) is an important synaptic dynamics that is capable of shaping the complex spatiotemporal activity of neural circuits. In this study, we examine the effects of STDP on the spatiotemporal patterns of a spatially extended, two-dimensional spiking neural circuit. We show that STDP can promote the formation of multiple, localized spiking wave patterns or multiple spike timing sequences in a broad parameter space of the neural circuit. Furthermore, we illustrate that the formation of these dynamic patterns is due to the interaction between the dynamics of ongoing patterns in the neural circuit and STDP. This interaction is analyzed by developing a simple model able to capture its essential dynamics, which give rise to symmetry breaking. This occurs in a fundamentally self-organizing manner, without fine-tuning of the system parameters. Moreover, we find that STDP provides a synaptic mechanism to learn the paths taken by spiking waves and modulate the dynamics of their interactions, enabling them to be regulated. This regulation mechanism has error-correcting properties. Our results therefore highlight the important roles played by STDP in facilitating the formation and regulation of spiking wave patterns that may have crucial functional roles in brain information processing.