Flash fiction - continuity or not?

Seeing Aldersprig write flash fiction with recurring characters/in firm settings (and seeing Ysabetwordsmith do something similar with poems) made me wonder about that.

As a reader, I like getting more stories about characters I like.

As a writer, I find working on one story in a series is a lot more difficult than writing a flash around one idea, because it only has to make sense within itself, rather than as part of a continuity.

In addition I wonder about the definition of flash fiction. As I understand it, a piece of flash is supposed to stand on its own. When does the label not fit anymore, and instead you have one chapter of a longer story that makes no sense on its own? Getting a short bit of fiction that is self-contained is the point of it, and using pre-existing characters and settings, I worry about not putting enough detail into the story for the "unfamiliar" reader, since with the things I know already about them, it makes sense with less.

Most of my flash fiction is absolutely one-off, built around an idea, with throwaway characters. That doesn't mean those stories don't require any context: Most of them draw on stereotypes or archetypes, genre conventions, and other things I assume people who read fantasy or science fiction to be familiar with. Then there's the fact that it's fanfiction in which drabbles (that is, flash fiction with exactly 100 words) first became popular, and while I'm not really a fanfiction writer, I imagine when you can put a fully developed character already known to your audience into your story just by naming them, the dynamic of writing is a bit different.

What do you think? What do you like to read, or write?