The movement patterns of a diverse range of animals have scale-free characteristics. These characteristics provide necessary but not sufficient conditions for the presence of movement patterns that can be approximated by Lévy walks. Nevertheless, it has been widely assumed that the occurrence of scale-free animal movements can indeed be attributed to the presence of Lévy walks. This is, in part, because it is known that the super-diffusive properties of Lévy walks can be advantageous in random search scenarios when searchers have little or no prior knowledge of target locations. However, fractional Brownian motions (fBms) and fractional Lévy motions (fLms) are both scale-free and super-diffusive, and so it is possible that these motions rather than Lévy walks underlie some or all occurrences of scale-free animal movement patterns. Here this possibility is examined in numerical simulations through a determination of the searching efficiencies of fBm and fLm searches. It is shown that these searches are less efficient than Lévy walk searches. This finding does not rule out the possibility that some animals with scale-free movement patterns are executing fBm and fLm searches, but it does make Lévy walk searches the more likely possibility.