(covers information from several alternate timelines)
This page contains information regarding Star Trek: Picard, and thus may contain spoilers.
Captain Jonathan Archer ordered the use of the long range sensors after Enterprise NX-01 went through the liquid phosphorus layer around a gas giant in 2151 and detected a Suliban helix. (ENT: "Broken Bow")
In August of 2151, the plasma cannon of the ECS Fortunate knocked out the long range sensors of Enterprise. When Trip Tucker was able to bring them back online, T'Pol detected weapon's fire and Enterprise went to settle the situation between the Fortunate and the Nausicaan pirates. (ENT: "Fortunate Son")
Also in 2151, Captain Archer ordered Lieutenant Malcolm Reed to stay on their long range scanners and let him know when the Enterprise made another contact with the unidentified aliens they had encountered earlier. (ENT: "Silent Enemy")
In 2367. Lieutenant Commander Geordi La Forge stated that the long range sensors aboard the USS Enterprise-D were able to scan a radius of ten light years within a twenty four hour period; effectively, one sector per day. (TNG: "The Wounded")
In 2368, the long range sensors aboard the Enterprise-D were able to detect a cubical Borg scout ship with a mass of 2.5 million metric tons, at a range that would take thirty-one hours and seven minutes to traverse at warp factor seven-point-six. (TNG: "I Borg").
In 2369, Lieutenant Commander Worf informed Commander William T. Riker that the long range sensors of the Enterprise-D picked up a distress call from a Romulan warbird. Later, La Forge was unable to find the Enterprise-D, as it was not at the rendezvous point. He told Captain Jean-Luc Picard that the long range sensors were at maximum and there was no sign of the Enterprise. (TNG: "Timescape")
The same year Doctor Julian Bashir and Major Kira Nerys responded to a priority one distress signal from the Kobliad transport Reyab and found the ship via the long range sensors. (DS9: "The Passenger")
During a rescue mission for the runabout USS Yangtzee Kiang in the Gamma Quadrant, the long range sensors of the USS Rio Grande were affected by the delta radiation of the star of a binary star system. (DS9: "Battle Lines")
In 2370, Sam Lavelle performed a long range scan to find an escape pod near the Cardassian border, but with no success. Data suggested it was possible that at this distance, the sensors of the Enterprise-D were unable to distinguish the pod from the inorganic matter normally present in space. (TNG: "Lower Decks")
Following the destruction of the Federation colony Ivor Prime near Deep Space 5 in 2373, long range sensors detected the Borg. Admiral Hayes reported this to Captain Jean-Luc Picard. (Star Trek: First Contact)