A: While one of the most obvious ways to get rid of a Creeper attached to one of your Keepers is to trash or move them both, there are some versions which include cards which let you trash or move the Creeper while still retaining the Keeper.

Cards that let you separate the two:
In Star Fluxx, and the various Star Trek Fluxxen, the Engineer (or version-specific named Engineer) can fix a Malfunctioning piece of equipment.
In Star Fluxx, the Doctor can cure Brain Parasites.
In Cthulhu Fluxx, the Sanitarium can cure Nightmares or Insanity.
In Anatomy Fluxx, It’s Contagious allows all players to detach and move Bacteria or Viruses to another player, while It’s Cured! allows

In Star Fluxx*, the exceptions are the Doctor, who can cure Brain Parasites, and the Engineer*, who can fix a Malfunction. In both cases they detach the Creeper, and the Keeper is left, good as new. Creeper Reassignment also specifically states that you detach the Creeper to move it. In Cthulhu Fluxx, the Dreamer states that he can detach Nightmares and discard it. Meanwhile the Sanitarium logically can cure an afflicted Keeper of Nightmares or Insanity. In Anatomy Fluxx, special actions allow you to “cure” yourself.

Some examples where the Creepers DO stay attached would be, in Star Fluxx, the weapons Laser Sword and Laser Pistol, which, again, quite logically, destroy the Keeper in order to destroy the attached Creeper. And while the Phaser in Star Trek or TNG Fluxx targets just the Creeper, it doesn’t say anything about detaching the Keeper either, so if you use it to get rid of an attached Creeper, the Keeper it’s attached to will also be destroyed. The Teleporter will move a Keeper… and its attached Creeper. If a Keeper or Creeper goes into the trash, then it’s attached card goes along.

In Cthulhu Fluxx, The Feds are pretty much the equivalent, though they also destroy themselves in the process. The Necronomicon lets you move any Creeper… and says nothing about detaching it, so you’d have to move any attendant Keeper (and extra Creepers if more than one is attached).

Trash Something is the generic Action version of what the weapons allow you to do. You could trash a Keeper, and it’s Creeper will go along, or you could trash a Creeper, and it’s attached Keeper would go along. There’s no logical reason or explicit wording that lets you detach connected Creeper-Keeper combos for this Action.

*In various Star Trek Fluxxen, there are specific Engineer analogues: Scotty, Geordi, O’Brien that work the same way with respect to Malfunction.