My impression in that most theories assume three spatial dimensions and one temporal dimension, though could in principle be formulated in others numbers of dimensions without inconsistencies. I know, however, that string theory makes specific predictions for the number of dimensions. It cannot be formulated in an arbitrary number of dimensions.
Is string theory unique in this regard? Are there other theories that predict (or even restrict) the numbers of dimensions? I'm interested in viable theories as well as toy models and historical ideas that turned out to be wrong.
I know for example that spontaneous symmetry breaking cannot occur in $d\le2$ spatial dimensions (Coleman/Mermin-Wagner theorem). Whilst it might be reasonable to expect that SSB is required in a theory of nature, I'm not sure whether I consider this to be a prediction for $d$ in QFT. I don't want to consider anthropic "predictions" a la Hawking.