The Supervision Field was the areas of the massive spaceship Red Dwarf, including mostly the human-inhabited areas (such as the Drive Room, sleeping quarters ect.), which were under the direct monitoring and control of the ship's computer Holly.
Within this field, Holly could detect the presence of non-human lifeforms, such as Frankenstein, and alert Captain Hollister to them. Lister was keeping his smuggled pet cat hidden in his sleeping quarters at the time. (The End, Series I)
Some areas of the ship, such as the cargo hold and Supply Pipe 28 (where the felis sapiens mostly evolved), were beyond the Supervision Field, and Holly was not able to see into these areas. The Cat would frequently disappear down Supply Pipe 28, and nobody, including Holly knew where to; Lister followed him down there once, finding the Cat Priest. (Waiting for God)
Rimmer and Lister, with the help of the Skutters, had to visually account for many supplies from the cargo hold, presumably since these were also beyond Holly's Supervision Field. Lister wasn't took keen on doing inventory checks, finding it boring, but Rimmer was controlling Lister as he had possession of Lister's cigarettes and insisted Lister to help him with them. (Balance of Power)
Holly later detected a tarantula-like creature aboard the ship, and warned Lister who had a terrible fear of such animals. In the Science Room, Lister felt it crawling up his leg and into his boxer shorts and, too scared to move, wrote a message to the Cat begging for help. It turned out that it was actually Kryten's hand and eyeball, sent in an ore sample pod to request help since Kryten had crashed on the Psi-moon. (Terrorform, Series V)
When there was a serious drought aboard the Red Dwarf, the internal scanners led them to the problem in the Water Tank on G Deck. This may have also contributed to the flooding of Holly's databanks. (Back to Earth)
Pree used the ship security system to watch and study the behavior of the crew, so that she could predict what their future behavior would be and complete actions in the way she knew the crew would have wanted without being asked. (Father & Suns, Series X)
As the various bio-printed Rimmer clones were being chased by the Rimmer Monster through the ship, the other Dwarfers watched a visual representation of this on the internal scanners, and which resembled Pac-Man. (Officer Rimmer, Series XI)
The Supervision Field presumably included recordings from the security surveillance system of the ship. It also presumably included some form of limited-range Psi-Scan, likely originating in - or near - the Drive Room, since the Drive Room was furthest area from the cargo bays - the latter area was where the field could not penetrate or see.