Picard being carried on a stretcher
A stretcher or gurney was a bed for carrying dead bodies or patients who required medical attention. Gurneys, in particular, had wheels to allow quick and easy movement without disturbing the patient, as well as straps to hold the patient in place during such movement.
In 2152, an injured and unconscious Hoshi Sato was carried to a gurney by a pair of Starfleet officers, on the Bridge of Enterprise NX-01. (ENT: "Minefield")
This bed was identified in the final draft script of "Minefield" as a gurney.
A short time later, a metallic stretcher was used to carry what was thought to be the corpse of Ensign Travis Mayweather out of launch bay 1 aboard Enterprise NX-01, on the way to the ship's sickbay. It was later discovered, though, that Mayweather was still alive and that the body was merely a fabrication. (ENT: "Dead Stop")
In January 2153, a team of three officers carried Hudak into Enterprise's sickbay on a stretcher, which they proceeded to place directly atop the room's main surgical bed, so he could be examined in the sickbay's imaging chamber. (ENT: "The Breach")
Later that year, T'Pol was carried on a makeshift stretcher, made from natural materials, by Jonathan Archer, Hoshi Sato, and Malcolm Reed. The stretcher was used on the Loque'eque homeworld, while T'Pol was unconscious and the other three were transformed into Loque'eque by a mutagenic virus. As soon as they realized she had regained consciousness, they put the stretcher down on the ground. T'Pol then realized she was actually bound to the stretcher, but she was soon released by the others, and they abandoned the stretcher. (ENT: "Extinction")
In an alternate 2259, an anti-grav equipped stretcher was seen at the Royal Children's Hospital. (Star Trek Into Darkness)
In 2367, an injured Jean-Luc Picard was transported by stretcher after he and Wesley Crusher were discovered following a crash landing on Lambda Paz. Doctor Beverly Crusher planned to have her son transported the same way, but Wesley assured her that he could walk. (TNG: "Final Mission")
Stretchers were used to transport the dead bodies of Cavit and Stadi off the bridge aboard the USS Voyager following the transport into the Delta Quadrant. (VOY: "Caretaker")
Harry Kim on a gurney
The Clown once placed Harry Kim on a gurney while forcing Kim to relive a traumatic childhood experience. (VOY: "The Thaw")
In 2373, multiple stretchers were used in the Battle of Ajilon Prime. Jake Sisko, at one point in the battle, offered to make a stretcher to carry an injured Burke. (DS9: "Nor the Battle to the Strong")
In scene 16 from the script for the Star Trek: The Next Generation first season episode "Encounter at Farpoint", the frozen Lieutenant Graham was being transported from the bridge on a floating stretcher. This scene was deleted from the aired episode.