The database currently contains all rational Sato-Tate groups of weight 1 and degree up to 4, as well as all Sato-Tate groups of weight 0 and degree 1 with component group of order at most $10^{20}$. Here are some further statistics.
 By weight: 0  1 By degree: 1  2  3  4 By identity component: $\mathrm{SO}(1)$   $\mathrm{U}(1)$   $\mathrm{SU}(2)$   $\mathrm{U}(1)_2$   $\mathrm{SU}(2)_2$   $\mathrm{U}(1)\times\mathrm{U}(1)$   $\mathrm{U}(1)\times\mathrm{SU}(2)$   $\mathrm{SU}(2)\times\mathrm{SU}(2)$   $\mathrm{USp}(4)$ Some interesting Sage-Tate groups or a random Sato-Tate group