I believe you're thinking of the all-versus-nothing proofs based on GHZ states. You start with a state such as
and you select at random one of the four measurements to implement: $X\otimes X\otimes X$, $X\otimes Y\otimes Y$, $Y\otimes X\otimes Y$ or $Y\otimes Y\otimes X$. Assuming every one of ...
Are there scenarios in which Bell nonlocality can be observed without such averages, that is, in a single-shot scenario?
No, you have to collect statistics. Any single result you see could have been due to classical players picking completely at random and getting lucky. Making the chance of classical luck arbitrarily close to zero requires repetition (or ...
Original paper: Going beyond Bell's theorem
Papers by Mermin:
Quantum mysteries revisited
Hidden variables and the two theorems of John Bell
TL;DR: We can prove that local hidden variables theories are incompatible with Quantum Mechanics without actually doing a measurement.
Neither collecting statistics no a single-shot measurement are needed.