Understanding the fine-scale connectivity structure of neural networks is key to deciphering their behaviour. Motifs in network connectivity are instrumental in relating circuit structure to dynamical properties of neural activity, such as covariability and dimensionality [1]. Experiments have made significant progress in measuring precise neural connectivity: multi-patch experiments, as well as more recent studies in connectomics, have found that certain types of three-neuron (triplet) connectivity motifs are over- or underrepresented relative to chance (Fig.1A, [2,3]). However, it is still unclear what type of synaptic plasticity mechanisms underlies the formation of these specific structures. Here, we investigated the emergence of motifs among three excitatory neurons using a family of spike-timing-dependent plasticity (STDP) rules, including Hebbian-like, anti-Hebbian-like, and symmetric rules, thus taking into account different types of spike-timing interactions (Fig.1B). We first developed an analytical framework to relate the parameters of the STDP rules and triplet connectivity structures, enabling us to predict, for any choice of STDP rule, which triplet motifs can emerge and which cannot (Fig.1C, D). We found that the ratio of long-term depression to long-term potentiation is the predominant factor in motif development, and that anti-Hebbian and symmetric STDP rules generate the same classes of motifs. Moreover, no single STDP rule is capable of generating the complete suite of triplet motifs found in neural data. Instead, a heterogeneous mixture of STDP rules is required to capture all triplet motifs observed in the data (Fig.E). All our analytical predictions are in perfect agreement with simulations of three spiking neurons as well as in larger networks, matching the small-scale structure results. In summary, our work unravels the effects of synaptic plasticity on triplet connectivity structures, demonstrating that a heterogeneous class of plasticity rules based on spike timing is sufficient to explain connectivity structures in neuronal circuits. These results establish an important foundation for understanding how covariability and dimensionality in neural representations emerge through learning.