A smooth cubic surface $X\subset \mathbb{P}^3$ is isomorphic to $\mathbb{P}^2$ blown up at six points, so there should be a birational map

$H^0(\mathbb{P}^3,\mathscr{O}_{\mathbb{P}^3}(3))//PGL_4\dashrightarrow {\rm Hilb}^6\mathbb{P}^2$

Given a 1-parameter family $X_t$ of smooth cubic surfaces specializing to $X_0$, how do the six points degenerate? For example, I know that $X_0$ having an $A_1$ singularity could be from 3 points becoming collinear or 6 points lying on a conic (contributing to a (-2) curve getting collapsed under the canonical map), but I don't know of references for other singularities.