With its substantial atmosphere of nitrogen, hydrocarbons and nitriles, Saturn's moon Titan is a unique planetary satellite. Photochemical processing of the gaseous constituents produces an extended haze that obscures the surface. Soon after the Voyager fly-bys in 1980 and 1981 photochemical models led to the conclusion that there should be enough liquid methane/ethane/nitrogen to cover the surface to a depth of several hundred metres. Recent Earth-based radar echoes imply that surface liquid may be present at a significant fraction of the locations sampled. Here we present ground-based observations (at near-infrared wavelengths) and calculations showing that there is no evidence thus far for surface liquid. Combined with the specular signatures from radar observations, we infer mechanisms that produce very flat solid surfaces, involving a substance that was liquid in the past but is not in liquid form at the locations we studied.