The theory of orderable groups is an old subject that has been extensively developed in the last few decades. I will describe various notions of orderability and how they relate to the dynamics of group actions on 1-dimensional manifolds. Thompson's groups occur naturally as important examples in this setting. I will introduce Polish topologies on various spaces of countable enumerated groups that satisfy one among various notions of orderability. Using elementary tools and examples from combinatorial group theory, combined with the Baire category theorem, I will describe how several phenomena are generic in these spaces. This is ongoing joint work with Goldbring and Kunnawalkam Ellayavalli.