  Two Magma functions are given: one computes linear systems of plane curves
with non-ordinary singularities and the other computes a scheme which
parametrizes given degree plane curves with given singularities. These
functions provide an efficient tool to construct explicit equations of singular
plane algebraic curves.
  By computing singular branch curves, we obtain equations of normal quartic
surfaces in $\mathbb C\mathbb P^3$ having the following combinations of
rational double points: $\mathsf D_5\mathsf E_7\mathsf E_7,$ $\mathsf
D_7\mathsf D_6\mathsf D_6,$ $\mathsf E_6\mathsf D_8\mathsf D_5,$ $\mathsf
E_6\mathsf D_{13},$ $\mathsf E_6\mathsf E_6\mathsf D_7,$ $\mathsf E_6\mathsf
E_8\mathsf D_5,$ $\mathsf E_7\mathsf D_6\mathsf D_6,$ $\mathsf E_7\mathsf
D_{12},$ $\mathsf E_7\mathsf E_6\mathsf D_6.$ These are all possible cases with
total Milnor number 19 which have no point of type $\mathsf A_n.$
