Hmm... here's a few off the top of my head.
1) A Vault full of teenage geniuses. Mainly the students from an elite private school (so under 18), with a few 'elders' (ie under 21) to fill the leadership roles. They think the 'true' Overseer, Doctor etc were lost in transit. Vault-Tek has made sure that their archives are well-stocked in cultural history. Purpose - they want to see if they can build their own society with no adults to guide them, and what culture they'd eventually develop.
2) A Vault where all the senior roles have been picked to have personalities that would clash. The Overseer would be weak-willed, and the other leaders would have their own agendas which would offend the others. The Vault has been deliberately designed to suffer from chronic minor problems to cause stress - and arguments on how to solve them. Purpose - would this community self-destruct, or could they learn to co-operate?
3) A minimally-functional Vault. The engineers had been working on the thing when the bombs fell, so they closed the door. The core functions are on-line, but they're missing much of the stuff to make it actually livable. On the plus side, they are Vault-Tek engineers, so if any group can jury-rig the place, it's them. Unfortunately, they still need stuff, which means venturing outside. Oh, and of the team of 35, only one is a woman.
4) The Vault with the insane quartermaster. On the surface, all looked well. Then as they started to root through the supplies, they realised that their supplies were a little... odd. They've got loads of stuff, but most 'everyday' items are critically short, while rarely-needed ones stocked to excess. (ie. no aspirin or underwear, but three crates of false teeth). The computer records and books are also weird; texts on advanced nuclear physics, but no Jack & Jane learning to read ones. Detailed maps of the entire US, with the exception of their local one. Purpose - could the Vault adapt all this non-standard kit so they survive?
