Herschel has shown that filamentary structures are ubiquitous in star-forming regions, in particular in nearby molecular clouds associated with Gould's Belt. High dynamic range far-infrared imaging of the Musca cloud with SPIRE and PACS reveals at least two types of filamentary structures: (1) the main ∼10-pc scale high column-density linear filament; and (2) low column-density striations in close proximity to the main filament. In addition, we find features with intermediate column densities (hair-like strands) that appear physically connected to the main filament. We present an analysis of this filamentary network traced by Herschel and explore its connection with the local magnetic field. We find that both the faint dust emission striations and the plane-of-the-sky (POS) magnetic field are locally oriented close to perpendicular to the high-density main filament (position angle ∼25−35 • ). The low-density striations and strands are oriented parallel to the POS magnetic field lines, which are derived previously from optical polarization measurements of background stars and more recently from Planck observations of dust polarized emission. The position angles are 97 ± 25 • , 105 ± 7 • , and 105 ± 5 • . From these observations, we propose a scenario in which local interstellar material in this cloud has condensed into a gravitationally-unstable filament (with "supercritical" mass per unit length) that is accreting background matter along field lines through the striations. We also compare the filamentary structure in Musca with what is seen in similar Herschel observations of the Taurus B211/3 filament system and find that there is significantly less substructure in the Musca main filament than in the B211/3 filament. We suggest that the Musca cloud may represent an earlier evolutionary stage in which the main filament has not yet accreted sufficient mass and energy to develop a multiple system of intertwined filamentary components.