The finite field Kakeya problem asks both the minimum size of a point set inAG(2, q)which contains a line in every direction, as well as a characterization of the examples. Blokhuis and Mazzocca  solved this problem, and a subsequent paper  addresses the stability of this solution for even order planes, i.e. the spectrum of sizes near the minimum size of a Kakeya set for which non-minimum Kakeya sets exist. In this paper we provide some computational results in small order planes to determine the full spectrum of sizes of Kakeya sets. We then address some spectrum issues on the upper end of possible sizes, providing some bounds and new constructions.We also address the question of minimality, i.e.whether a given Kakeya set contains any smaller Kakeya set.