Abstract: We consider a transport equation by a gradient vector field with a small viscous perturbation. We study uniform observability properties from a small subset in the (singular) vanishing viscosity limit. We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation. We also prove that the two minimal times coincide for positive solutions. This is a joint work with Camille Laurent.