We study one-mode Gaussian quantum channels in continuous-variable systems by performing a black-box characterization using complete positivity and trace preserving conditions, and report the existence of two subsets that do not have a functional Gaussian form. Our study covers as a particular limit the case of singular channels, thus connecting our results with their known classification scheme based on canonical forms. Our full characterization of Gaussian channels without Gaussian functional form is completed by showing how Gaussian states are transformed under these operations, and by deriving the conditions for the existence of master equations for the non-singular cases. We show that although every functional form can be found in the vicinity of the identity, one of them does not parametrize unitary channels.