Flat optics nanogratings supported on thin free-standing membranes offer the opportunity to combine narrowband waveguided modes and Rayleigh anomalies for sensitive and tunable biosensing. At the surface of high-refractive index Si3N4 membranes we engineered lithographic nanogratings based on plasmonic nanostripes, demonstrating the excitation of sharp waveguided modes and lattice resonances. We achieved fine tuning of these optical modes over a broadband Visible and Near-Infrared spectrum, in full agreement with numerical calculations. This possibility allowed us to select sharp waveguided modes supporting strong near-field amplification, extending for hundreds of nanometres out of the grating and enabling versatile biosensing applications. We demonstrate the potential of this flat-optics platform by devising a proof-of-concept nanofluidic refractive index sensor exploiting the long-range waveguided mode operating at the sub-picoliter scale. This free-standing device configuration, that could be further engineered at the nanoscale, highlights the strong potential of flat-optics nanoarrays in optofluidics and nanofluidic biosensing.