  Motivated by the search for rational points in moduli spaces of
two-dimensional conformal field theories, we investigate how points with
enhanced symmetry algebras are distributed there. We first study the bosonic
sigma-model with $S^1$ target space in detail and uncover hitherto unknown
features. We find for instance that the vanishing of the twist gap, though true
for the $S^1$ example, does not automatically follow from enhanced symmetry
points being dense in the moduli space. We then explore the supersymmetric
sigma-model on K3 by perturbing away from the torus orbifold locus. Though we
do not reach a definite conclusion on the distribution of enhanced symmetry
points in the K3 moduli space, we make several observations on how chiral
currents can emerge and disappear under conformal perturbation theory.
