We investigate the stability of continuous-time quantum walks (CTQW) across cycle, complete, star, Erdős-Rényi, small-world, and scale-free topologies under energy-based intrinsic decoherence, node-based Haken-Strobl noise, and edge-based quantum stochastic walk (QSW) decoherence. Defining stability as the preservation of quantum properties, we characterize it using node probabilities, $\ell_1$-norm of coherence, fidelity, quantum-classical distance, and von Neumann entropy. Our results show that intrinsic decoherence preserves coherence longest while QSW causes rapid decay. Stability rankings vary and depend on the decoherence types, network structure, and properties of node where the walker is initialized specifically in heterogeneous networks. Dense connected network like complete and heterogenous networks, for instance, star, and scale-free are stable under Haken-Strobl noise but become uniquely fragile under QSW when initialized on high degree nodes. However, these same networks, due to their inherent localization, exhibit lower coherence in the noiseless regime, highlighting a fundamental trade-off between localization and coherence. Furthermore, the centrality of the initialization node has a pronounced impact on relaxation time and stability measures, underscoring the critical role of local topological features in quantum dynamics.