Plasticity rules governing synaptic changes vary widely across species, brain regions, and brain states. Despite this variability, these rules often give rise to neural circuits with similar structural motifs and functional properties, a phenomenon known as degeneracy. While functional degeneracy---where different plasticity rules result in similar network functions---has been widely studied, the structural aspects of degeneracy remain underexplored. Structural degeneracy refers to cases where distinct plasticity rules produce neural circuits with similar connectivity patterns. To investigate emergent structural properties, we performed a comparative analysis of the relationship between plasticity rules and their resulting network structures. Using filter simulation-based inference (fSBI) to generate a large dataset of spike timing-dependent plasticity (STDP) rules optimized for stabilizing recurrent spiking neural networks (Confavreux et al., 2023), we examined how these rules shape network architectures over time. Our findings confirm that distinct plasticity rules can give rise to highly similar network structures, exemplifying structural degeneracy. In contrast, we also identified a phenomenon we term fragility, where nearly identical plasticity rules lead to drastically different network structures. These two phenomena---degeneracy and fragility---highlight the diverse and complex interplay between plasticity rules and network architectures, offering new insights into the mechanisms underlying synaptic plasticity.