The adsorption of a single chain on structured surfaces is investigated using the bond-fluctuation model. The different surfaces were a flat, a stick-like, a pyramidal surface and a surface with square holes. The adsorption starts at about the same temperature for all surface structures, therefore denoted as the adsorption temperature. The rate for further adsorption with decreasing temperature is enhanced for the structured surfaces. The desorption of once adsorbed chains starts markedly below the adsorption temperature. Despite the structure of the surfaces the adsorbed chains show clearly two-dimensional properties for the lowest temperatures. The exception is the surface with square holes. Surprisingly, the chains on this surface keep their three-dimensional properties even when adsorbed onto the surface.