We study the electronic eigenstates on two- and three-dimensional quasiperiodic lattices using a tight-binding Hamiltonian in the vertex model. In particular, we analyse how a quasiperiodic lattice influences the decay form and the self-similar structure of the wave functions. The investigation of the earlier-suggested power-law localization is performed by calculating participation numbers and the structural entropy of the wave function. We also present results for the multifractal analysis of the eigenstates by a standard box-counting method. The eigenstates of the two-dimensional Penrose lattice display multifractal character. In contrast, most eigenstates of the three-dimensional Amman-Kramer lattice are shown to be extended; localized states occur only in the band tails, where the spectrum appears to be fractal.