Over at Does every set admit a rigid binary relation?, I showed that at least for sets of reals, there is an affirmative answer. Namely, every set of reals admits a rigid binary relation, by an argument that uses neither the Axiom of Choice nor Foundation (and does not make use of any hereditary well-founded structure of the reals, so it would apply even if you conceived of numbers as urelements).

But it is not clear to me how to generalize this to higher sets.

Over at Does every set admit a rigid binary relation?, I showed that at least for sets of reals, there is an affirmative answer. Namely, every set of reals admits a rigid binary relation, by an argument that uses neither the Axiom of Choice nor Foundation (and does not make use of any hereditary well-founded structure of the reals, so it would apply even if you conceived of numbers as urelements).

But it is not clear to me how to generalize this to higher sets.