Sector 010 was a region of space. A portion of the Romulan Neutral Zone was located in sector 010/35.
In 2364, the Federation starships USS Tranquillity Base, USS Non Sequitur, and USS Max Plank were ordered by Starfleet Command to decrease sensor drone surveillance of this sector.(TNG: "Conspiracy", display graphic)
A year later, NCC-31054 was assigned a deep exploration mission in this sector. (TNG: "The Measure Of A Man")
The sector was never mentioned in dialog; it was mentioned on mission orders and, later, on a Starship Deploy Status.