A day-ahead voltage-stability-constrained network topology optimization (DVNTO) problem is proposed to find the day-ahead topology schemes with the minimum number of operations (including line switching and bus-bar splitting) while ensuring the sufficient hourly voltage stability margin and the engineering operation requirement of power systems. The AC continuation power flow and the uncertainty from both renewable energy sources and loads are incorporated into the formulation. The proposed DVNTO problem is a stochastic, large-scale, nonlinear integer programming problem. To solve it tractably, a tailored three-stage solution methodology, including a scenario generation and reduction stage, a dynamic period partition stage, and a topology identification stage, is presented. First, to address the challenges posed by uncertainties, a novel problem-specified scenario reduction process is proposed to obtain the representative scenarios. Then, to obtain the minimum number of necessary operations to alter the network topologies for the next 24-hour horizon, a dynamic period partition strategy is presented to partition the hours into several periods according to the hourly voltage information based on the voltage stability problem. Finally, a topology identification stage is performed to identify the final network topology scheme. The effectiveness and robustness of the proposed three-stage solution methodology under different loading conditions and the effectiveness of the proposed partition strategy are evaluated on the IEEE 118-bus and 3120-bus power systems.