One useful case that has arisen in my own work is when $lim_{t\to 0} t\cdot x$ exists for all $x\in X$. This sort of action is "circle compact" and then the BB decomposition goes through. I think the terminology is due to Tamas Hausel -- in the case where $X$ is smooth and quasi-projective, this condition is the same as saying that the moment map for the $S^1$ action is proper.

One useful case that has arisen in my own work is when $lim_{t\to 0} t\cdot x$ exists for all $x\in X$. This sort of action is "circle compact" and then the BB decomposition goes through. I think the terminology is due to Tamas Hausel -- in the case where $X$ is smooth and quasi-projective, this condition is the same as saying that the moment map for the $S^1$ action is proper and bounded below.

(Proper isn't enough, as the example of $S^1$ acting on {$(x,y) : xy=1$} shows.)