Let us have a (possibly singular) irreducible projective variety $X$ over $\mathbb{C}$, with an algebraic $\mathbb{C}^*$-action that has finitely many fixed points $\{x_1,\dotsc,x_n\}$. One can define the attracting sets $$U_k = \{x \in X \mathrel| \lim_{t\rightarrow \infty}t\cdot x =x_k\}$$ that decompose $X$ into a disjoint union. When $X$ is smooth, Białynicki-Birula's theorem is that these are affine bundles over $x_k$, hence affine spaces. That gives us an affine cell decomposition of $X$.

Is this still true when $X$ is singular?

someresults, just not as nice, e.g., decomposition is not into affine spaces; but maybe it is still useful for you? For instance, arxiv.org/abs/1308.2604 is for $G_m$ action on algebraic spaces of finite type; doi.org/10.1016/j.matpur.2019.04.006 generalizes it to actions of reductive groups. $\endgroup$4more comments