Every finite-dimensional irreducible representation of a (classical) affine Lie algebra has quantum analogues, but these are generally ’larger’ than their classical counterparts. Among the quantum analogues of a particular classical representation, some (usually one) are ’minimal’ in a certain precise sense. This paper studies the structure of these minimal representations when the underlying finite-dimensional Lie algebra is of rank 2. We also compute their q-characters in some cases.