Improving the cooling performance of heat sink is critical for providing better thermal management to compact electronic devices. Thus, analyzing various cavity shapes for a heat sink containing Nano-enhanced Phase Change Material is of utmost necessity. The present study focuses on a two-dimensional conjugate heat propagation in a Heat sink with Rectangular (HRC), Trapezoidal (HTC), and Wavy (HWC) cavities. Various cavity shapes are examined to identify the ideal shape for a heat sink containing nano-enhanced Phase Change Material (Ne-PCM). The copper oxide (CuO) nanoparticle enhanced paraffin is considered as Ne-PCM. The analysis encompasses identification of the potent cavity shape, its optimum geometric parameters, the dominant mode of heat transfer and ultimately the desirable thermophysical properties of the Ne-PCM. Among HRC, HTC, and HWC, the heat sink with HRC performs the best by maintaining the lowest base temperature. The HTC, has a better initial performance than the HRC, though HTC has low PCM content. Nevertheless, as time progresses, HTC fails to inhibit the base temperature rise. In contrast, due to increased contact surface area, PCM in HWC experiences a stronger opposing viscous force. Consequently, HWC has conduction-dominated heat transfer and the slowest melting of PCM that elevates the base temperature, accumulating heat near cavity walls. Shortening the relative pitch distance and elongating the relative height of the HRC further lowers the base temperature of the heat sink. Notably, the heat transfer mode analysis reveals that in the cavity with maximum height, heat diffusion dominates convective heat transfer. Heat diffusion increases away from the cavity walls towards the middle of the cavity. Ultimately, it is noted that a denser PCM with high thermal conductivity and low viscosity effectively maintains a low heat sink base temperature. A denser PCM imparts more sensible and latent heat absorption capability and maintains a lower base temperature. On complete melting, the convective heat transfer is much higher in a denser PCM. A PCM with higher thermal conductivity marginally lowers the base temperature with a small increase in PCM melting rate.