Multigraft polymers comprise a subclass of branched polymers where more than one side chain is attached to each node (branching point) of the main chain. We have investigated structural properties of single multigraft polymers under good solvent conditions by Monte Carlo simulations, employing a flexible bead-spring model. Beside the grafting density, denoting the linear density of grafted side chains, we have introduced the concept of branching density, denoting the linear density of nodes. At high branching density, both the branching density and the branching multiplicity controlled the structure of the side chains, whereas at lower branching density only the branching multiplicity influenced the side-chain structure. The spatial extension of the main chain and side chains as a function of side-chain length and grafting density was analyzed using scaling formalism. The dependence of the main-chain extension on side-chain length, branching density, and branching multiplicity could be collapsed on a universal curve upon relevant rescaling. Multigraft polymers with equal number of side-chain beads but unequal numbers and lengths of side chains displayed unconventional bending properties. Few and long side chains gave rise to a still relative low locally stiffness but considerable long-range rigidity, whereas more numerous and shorter side chains lead to a higher local stiffness but to a smaller long-range rigidity.