Context. Small-scale magnetic flux emergence in the quiet Sun is crucial for maintaining solar magnetic activity. On the smallest scales studied so far, namely within individual granules, two mechanisms have been identified: emergence in tiny magnetic loops and emergence in the form of magnetic flux sheets covering the granule. While there are abundant observations of tiny magnetic loops within granules, the evidence for the emergence of granule-covering magnetic sheets is much more limited. Aims. This work aims to statistically analyse magnetic flux sheets, quantify their frequency on the solar surface and their potential contribution to the solar magnetic budget in the photosphere, and investigate the plasma dynamics and granular-scale phenomena associated with their emergence. Methods. Using spectro-polarimetric datasets taken along the Fe I 630.15 and 630.25 nm photospheric lines from the solar optical telescope aboard the Hinode satellite and the Fe I 630.15, 630.25 nm, and 617.3 nm ones from the ground-based Swedish Solar telescope, we developed a two-step method to identify magnetic flux sheet emergence events, detecting magnetic flux patches based on the calculation of the transverse and longitudinal magnetic flux density and associating them with their host granules based on velocity field analysis. Results. We identified 42 events of magnetic flux sheet emergence and characterised their magnetic properties and the associated plasma dynamics of their host granules. Our results align with numerical simulations, indicating a similar occurrence rate of approximately 0.3 events per day per Mm2. We investigated the relationship between magnetic flux emergence and granular phenomena, finding that flux sheets often emerge in association with standard nascent granules, as well as exploding granules, or granules with granular lanes. In particular, we highlight the potential role of recycled magnetic flux from downflow regions in facilitating flux sheet emergence. Our analysis suggests that the magnetic flux sheet events could be considered part of the larger component of the distribution of small-scale magnetic flux that feeds the solar atmosphere in quiet Sun regions.