Let $X\subset \mathbb{P}^3$ be an irreducible quartic surface, defined over an algebraically closed field $k$. Suppose that $X$ is rational (i.e. birational to $\mathbb{P}^2$). Is is true that $X$ has one of the following type of singularities?

a) a point $q\in X$ of multiplicity $4$ (hence $X$ is a cone over a rational quartic curve).

b) a point $q\in X$ of multiplicity $3$ (hence the projection away from $q$ gives a birational map $X\dashrightarrow \mathbb{P}^2$).

c) a curve of singular points.

d) a double point $q\in X$ and a double line $l$ in the exceptional divisor $E_q\simeq \mathbb{P}^2$ of $q$ **edit: new case added**

e) a double point $q\in X$, a double point $q_1$ in the exceptional divisor $E_q\simeq \mathbb{P}^2$ of $q$ and a double line $l$ in the exceptional divisor $E_{q_1}\simeq \mathbb{P}^2$ of $q_1$ **edit: new case added**

It seems from the classification of Noether indicated by abx in 

> M. Noether, "Ueber die rationalen Flächen vierter Ordnung" (Mathematische Annalen, 1189, page 546-571)

that these are the only possible cases. Is there a modern proof of this ?