Semiflexible polymers end-grafted to a repulsive planar substrate under good solvent conditions are studied by scaling arguments, computer simulations, and self-consistent field theory. Varying the chain length N, persistence length lp, and grafting density σg, the chain linear dimensions and distribution functions of all monomers and of the free chain ends are studied. Particular attention is paid to the limit of very small σg, where the grafted chains behave as "mushrooms" no longer interacting with each other. Unlike a flexible mushroom, which has a self-similar structure from the size (a) of an effective monomer up to the mushroom height (h/a ∝ N(v), ν ≈ 3/5), a semiflexible mushroom (like a free semiflexible chain) exhibits three different scaling regimes, h/a ∝ N for contour length L = Na < lp, a Gaussian regime, h/a ∝ (Llp)(1/2)/a for lp ≪ L ≪ R* ∝ (lp(2)/a), and a regime controlled by excluded volume, h/a ∝ (lp/a)(1/5)N(ν). The semiflexible brush is predicted to scale as h/a ∝ (lpaσg)(1/3)N in the excluded volume regime, and h/a ∝ (lpa(3)σ(2))(1/4)N in the Gaussian regime. Since in the volume taken by a semiflexible mushroom excluded-volume interactions are much weaker in comparison to a flexible mushroom, there occurs an additional regime where semiflexible mushrooms overlap without significant chain stretching. Moreover, since the size of a semiflexible mushroom is much larger than the size of a flexible mushroom with the same N, the crossover from mushroom to brush behavior is predicted to take place at much smaller densities than for fully flexible chains. The numerical results, however, confirm the scaling predictions only qualitatively; for chain lengths that are relevant for experiments, often intermediate effective exponents are observed due to extended crossovers.