We investigate the dependence on the search space dimension of statistical properties of random searches with Lévy α-stable and power-law distributions of step lengths. We find that the probabilities to return to the last target found (P0 ) and to encounter faraway targets (PL ), as well as the associated Shannon entropy S, behave as a function of α quite differently in one (1D) and two (2D) dimensions, a somewhat surprising result not reported until now. While in 1D one always has P0 PL, an interesting crossover takes place in 2D that separates the search regimes with P0 > PL for higher α and P0 < PL for lower α, depending on the initial distance to the last target found. We also obtain in 2D a maximum in the entropy S for α ∈ (0, 2], not observed in 1D apart from the trivial α → 0 ballistic limit. Improving the understanding of the role of dimensionality in random searches is relevant in diverse contexts, as in the problem of encounter rates in biology and ecology.