I will present some aspects of an ongoing project with Vadim Alekseev, where we study the structure of amenable subgroups of arithmetic lattices from different point of views. We will show that the structure of the ambient algebraic group gives us a way of understanding the amenable subgroups up to commensurability, that is, up to passing to finite index subgroups. We will give several explicit examples, we will show some connections to some (basic) number theory and how one can design algorithms to find these amenable subgroups. Finally we will discuss about the notion of singularity, why some of these amenable subgroups are singular and how it is connected to ergodic theory and von Neumann algebras.