Amputation was a surgical procedure whereby a patient's appendages were removed. It could be performed due to medical necessity or as a punishment. A part of the body that had been traumatically amputated in an accident could be said to have been severed. (DS9: "Nor the Battle to the Strong", "Soldiers of the Empire"
A deleted scene in "Broken Bow" had Reed jokingly tell Hoshi Sato, who believed she suffered from frostbite, that Dr. Phlox might have to amputate.
In 2151, Trip Tucker noted that if one were to rest one's hands on the handrail of a lift in Enterprise NX-01's engineering section during the lift's operation, one's fingers might be amputated. (ENT: "Unexpected")
In 2152, following his impalement by a Romulan mine, Malcolm Reed joked that he might consider allowing his leg to be amputated, except he thought that if Chef got hold of the severed limb, he would serve "Roast Reed" next Sunday. (ENT: "Minefield")
Later that year, when Tucker assured Kaitaama that he wouldn't attempt to woo her, she warned him that if he did, the Sovereign Guard would cut off his hand. Tucker then quipped, "You must be a fun date." (ENT: "Precious Cargo")
In 2369, William T. Riker's arm was amputated and then surgically re-attached by solanogen-based lifeforms that were experimenting on crew members of USS Enterprise-D. (TNG: "Schisms")
In 2375, a severe injury to Nog's left leg necessitated amputation after Nog was shot by Jem'Hadar on AR-558 (DS9: "The Siege of AR-558"), requiring him to deal with some psychological issues relating to the pain he felt even after receiving a new synthetic limb in its place. Vic Fontaine was able to help him with this psychological pain. (DS9: "It's Only a Paper Moon")
Also that year, when Tomin touched Seven of Nine's arm when inviting her to return to his quarters, him having studied Human mating rituals, she told him to "Remove your hand or I will remove your arm." (VOY: "Someone to Watch Over Me")
In 2376, Lieutenant Maxwell Burke joked with Seven of Nine about needing amputation for her minor injuries (first degree phaser burns and minor lacerations), which Seven did not appreciate. (VOY: "Equinox, Part II")