  We prove upper and lower bounds on the minimal spherical dispersion,
improving upon previous estimates obtained by Rote and Tichy [Spherical
dispersion with an application to polygonal approximation of curves, Anz.
\"Osterreich. Akad. Wiss. Math.-Natur. Kl. 132 (1995), 3--10]. In particular,
we see that the inverse $N(\varepsilon,d)$ of the minimal spherical dispersion
is, for fixed $\varepsilon>0$, linear in the dimension $d$ of the ambient
space. We also derive upper and lower bounds on the expected dispersion for
points chosen independently and uniformly at random from the Euclidean unit
sphere. In terms of the corresponding inverse $\widetilde{N}(\varepsilon,d)$,
our bounds are optimal with respect to the dependence on $\varepsilon$.
