Polymers grafted with one chain end to an impenetrable flat hard wall which attracts the monomers with a short-range adsorption potential (of strength epsilon) are studied by large scale Monte Carlo simulations, using the pruned-enriched Rosenbluth method (PERM). Chain lengths up to N = 25600 steps are considered, and the intrinsic flexibility of the chain is varied via an energy penalty for nonzero bond angles, epsilon(b). Choosing q(b) = exp(-epsilon(b)/k(B)T) in the range from q(b) = 1 (fully flexible chains) to q(b) = 0.005 (rather stiff chains with a persistence length of about l(p) = 52 lattice spacings), the adsorption transition is found to vary from about epsilon/k(B)T(c) approximate to 0.286 to epsilon/k(B)T(c) approximate to 0.011, confirming the theoretical expectation that epsilon/k(B)T(c) proportional to 1/l(p) for large l(p). The simulation data are compatible with a continuous adsorption transition for all finite values of l(p), while in the rigid rod limit (l(p) -> infinity) a first order transition seems to emerge. Scaling predictions and blob concepts on the structure of weakly adsorbed semiflexible polymers absorbed at interfaces are briefly discussed.