Of course there are the $p$-subgroups of a maximal torus, and in the case $G=PU_p$, there is an interesting non toral elementary p-subgroup considered by Vistoli in this paper.

How many other cases are known? For example, how about $G=PU_n$ where $n$ is not a prime?

How to determine (say up to conjugacy) elementary $p$-subgroups of a compact Lie group $G$?

Of course there are the $p$-subgroups of a maximal torus, and in the case $G=\mathrm{PU}_p$, there is an interesting non-toral elementary $p$-subgroup considered by Vistoli in this paper.

How many other cases are known? For example, how about $G=PU_n$ where $n$ is not a prime?