Gathering is a fundamental coordination problem in cooperative mobile robotics. In short, given a set of robots with \emph{arbitrary} initial locations and no initial agreement on a global coordinate system, gathering requires that all robots, following their algorithm, reach the exact same but not predetermined location. Gathering is particularly challenging in networks where robots are oblivious (i.e.,~stateless) and direct communication is replaced by observations on their respective locations. Interestingly any algorithm that solves gathering with oblivious robots is inherently self-stabilizing if no assumption is made on the initial distribution of the robots. In this paper, we significantly extend the studies of deterministic gathering feasibility under different assumptions related to synchrony and faults (crash and Byzantine). Unlike prior work, we consider a larger set of scheduling strategies, such as bounded schedulers, and derive interesting lower bounds on these schedulers. In addition, we extend our study to the feasibility of probabilistic self-stabilizing gathering in both fault-free and fault-prone environments.