The deployment of AI on edge computing devices faces significant challenges related to energy consumption and functionality. These devices could greatly benefit from brain-inspired learning mechanisms, allowing for real-time adaptation while using low-power. In-memory computing with nanoscale resistive memories may play a crucial role in enabling the execution of AI workloads on these edge devices. In this study, we introduce voltage-dependent synaptic plasticity (VDSP) as an efficient approach for unsupervised and local learning in memristive synapses based on Hebbian principles. This method enables online learning without requiring complex pulse-shaping circuits typically necessary for spike-timing-dependent plasticity (STDP). We show how VDSP can be advantageously adapted to three types of memristive devices (TiO$_2$, HfO$_2$-based metal-oxide filamentary synapses, and HfZrO$_4$-based ferroelectric tunnel junctions (FTJ)) with disctinctive switching characteristics. System-level simulations of spiking neural networks incorporating these devices were conducted to validate unsupervised learning on MNIST-based pattern recognition tasks, achieving state-of-the-art performance. The results demonstrated over 83% accuracy across all devices using 200 neurons. Additionally, we assessed the impact of device variability, such as switching thresholds and ratios between high and low resistance state levels, and proposed mitigation strategies to enhance robustness.