The Federation Science Council was a group of officials of the United Federation of Planets that advised the Federation Council on matters pertaining to Federation security, in regards to scientific aspects.
A sign in Doctor Paul Manheim's laboratory on Vandor IV stated that Federation Science Council Regulation 5244 had to be observed at all times when the facility was in use, further stating that all operational equipment had to be rated for twenty-four mili-cochranes or greater. (TNG: "We'll Always Have Paris")
In 2370, Lieutenant Sanders of the science station on Calder II told Commander Riker that he could not allow anyone to enter the ruins on Calder II without prior authorization by the Federation Science Council. (TNG: "Gambit, Part I")
A few years before 2370, Hekaran scientists Serova and Rabal had repeatedly explained to the Federation Science Council that warp drive was damaging subspace, specifically in and near the Hekaras Corridor near their homeworld. The Council, knowing how much warp drive meant to Starfleet and Federation security, didn't believe their theory had enough evidence, and dismissed them.
However, in 2370, when Serova committed suicide to open a subspace rupture with her shuttle's warp core, she proved her theory was true. To prevent this from happening elsewhere, not merely the Science Council, but the Federation Council itself implemented a warp 5 "speed limit" throughout areas prone to these fractures, until a way to stop it could be developed. (TNG: "Force of Nature")