Natural gas (methane, CH4) is widely considered as a promising energy carrier for mobile applications. Maximizing the storage capacity is the primary goal for the design of future storage media. Here we report the CH4 storage properties in a family of isostructural (3,24)-connected porous materials, MFM-112a, MFM-115a, and MFM-132a, with different linker backbone functionalization. Both MFM-112a and MFM-115a show excellent CH4 uptakes of 236 and 256 cm3 (STP) cm-3 (v/v) at 80 bar and room temperature, respectively. Significantly, MFM-115a displays an exceptionally high deliverable CH4 capacity of 208 v/v between 5 and 80 bar at room temperature, making it among the best performing metal-organic frameworks for CH4 storage. We also synthesized the partially deuterated versions of the above materials and applied solid-state 2H NMR spectroscopy to show that these three frameworks contain molecular rotors that exhibit motion in fast, medium, and slow regimes, respectively. In situ neutron powder diffraction studies on the binding sites for CD4 within MFM-132a and MFM-115a reveal that the primary binding site is located within the small pocket enclosed by the [(Cu2)3(isophthalate)3] window and three anthracene/phenyl panels. The open Cu(II) sites are the secondary/tertiary adsorption sites in these structures. Thus, we obtained direct experimental evidence showing that a tight cavity can generate a stronger binding affinity to gas molecules than open metal sites. Solid-state 2H NMR spectroscopy and neutron diffraction studies reveal that it is the combination of optimal molecular dynamics, pore geometry and size, and favorable binding sites that leads to the exceptional and different methane uptakes in these materials.