We analyze the structural behavior of isolated semiflexible polymers grafted to an impenetrable surface. Employing a parallel multicanonical algorithm, we find a rich phase behavior for both purely entropic and attractive surface-polymer interactions. The corresponding conformations range from very compact ones, to folded bundles, to adsorbed and desorbed weakly bent rods. The case of a polymer grafted to a flat, noninteracting surface differs only marginally from the case of a completely free and isolated polymer. Introducing surface attraction, we find that below the adsorption transition temperature stiffer polymers are completely adsorbed and confined to an effectively two-dimensional conformation space. This ultimately holds true over the full range of semiflexibility for increasing surface attraction, in accordance with recent experimental findings.