Binary neutron-star (BNS) mergers are accompanied by multi-messenger emissions, including gravitational wave (GW), neutrino, and electromagnetic signals. Some fraction of BNS mergers may result in a rapidly spinning magnetar as a remnant, which can enhance both the EM and neutrino emissions. In the first part of this talk, I will discuss the possible neutrino signatures from such systems. In the second part, I will discuss the possibility and prospects of performing GW-triggered stacking searches for high-energy and ultra-high energy neutrinos from BNS mergers in general, at IceCube-Gen2, GRAND, and RNO-G, using the next generation of GW detectors.