Which countable discrete groups (apart from the infinite amenable ones) admit uncountably many mutually non-conjugate free ergodic probability measure preserving actions that are all mutually orbit equivalent?
We consider infinite countable groups and ask about the this property: admitting uncountably many mutually non-isomorphis free ergodic probability measure preserving (pmp) actions that are all mutually orbit equivalent.
You ask which groups satisfy this property. I do not know exactly. I think no one knows. But it is a big class of groups, maybe all groups which do not satisfy property (T).
In Corollary 2.6 of A converse to Dye’s theorem Hjorth proves that property (T) groups do not satisfy this property.
On the other hand, amenable groups do satisfy this property. Indeed, any amenable group has uncountably many mutually non-isomorphic pmp actions (say, by entropy theory), but these are all orbit equivalent, by the celebrated Ornstein-Weiss theorem.
To see that there are many non-amenable groups satisfying this property, fix a pmp action of a non-amenable group $\Gamma_1$ on $X$ and uncountably many pmp actions of an amenable group $\Gamma_2$ on $Y_\alpha$. Then the non-amenable group $\Gamma=\Gamma_1\times \Gamma_2$ will satisfy this property, as witness by its actions on $X\times Y_\alpha$.
If you want to study this property further, I suggest you to start with the following two things: try to show that if a group has a quotient that satisfies this property then the group satisfies this property, and try to show that this property is a measure equivalence invariant.