A displacement gun was, according to Miss Quill, one of the few weapons in time and space able to kill Shadow Kin. It did this by hitting both target and shooter, tearing the body apart in the process, and scattering it to the far corners of the universe, taking the Shadow Kin with it. Quill had one, but she couldn't use it, due to the arn in her head preventing her from causing physical harm.
She nevertheless tricked Kevin Williams into using it to destroy both himself and a Shadow Kin, and nearly succeeded in doing the same thing to April MacLean, were Charlie not to push her out of the way at the last second. Afterwards Miss Quill used it to scan April to discover that she and Corakinus shared one heart.
At the prom, Charlie used it to ward off Shadow Kin; in this setting, the gun did not shoot both ways. Corakinus stepped on the gun, destroying it.
Later, the Lankin tried to entice Quill with a fully repaired displacement gun in order to persuade her into physical contact, which would allow the Lankin to consume her grief and thus kill her in the process. She later vowed to repair the original gun, taking its broken remains from a drawer. (TV: Nightvisiting)
Dorothea Ames gave Quill and Ballon a newly repaired gun to fight to the death with so one of them could escape the Cabinet of Souls. Ballon took aim at Quill, but it backfired, killing him. Quill then took the gun. (TV: The Metaphysical Engine, or What Quill Did)