Forums like Questionablequesting just use your run-of-the-mill forum software a la phpBB maybe with a dice rolling function. Anonkun was developed exclusively to facilitate quest threads, pic related. They have tags, live chat, built-in polling, etc. I can't claim to spend much or any time there, but it seems very extensive.

Meanwhile, running quests on 4chan is a little bit of a hassle. There's no way to eliminate ballot-stuffing ("samefagging"), no ability to continue threads (see also: /vg/ thread #1937 about the same topic), and no polling function. Discussion seems to be the only way to go, but that's impossible with large groups (as 4chan is not a chatroom; see any thread with many concurrent participants).

Quest threads as defined above would likely include Risk threads: they are competitive, however, and do not tell a story. The only original content these threads produce is maps filled in differently with a paint-bucket tool. I am content to include these in the definition, but perhaps they are worth discussing separately.

There are currently quests on /tg/, /u/ ("CYOA"), and, recently, /d/.

There have also been quests on /a/, /co/, and /v/. I use the past tense, because quests have been driven off these boards with either a combination of users spamming the threads, or janitors/moderators deleting threads as they are happening, see OP pic.

There doesn't seem to be a consistent understanding about where quests go. I've heard it said before that they are OK on /co/, but users don't seem to agree, remembering the purging of Teen Titans quest.

TL;DR: Where are quests OK, and where are they not OK? The current situation is confusing.