  Plane quartics containing the ten vertices of a complete pentalateral and
limits of them are called L\"uroth quartics. The locus of singular L\"uroth
quartics has two irreducible components, both of codimension two in $\P^{14}$.
We compute the degree of them and we discuss the consequences of this
computation on the explicit form of the L\"uroth invariant. One important tool
are the Cremona hexahedral equations of the cubic surface. We also compute the
class in $\overline{M}_3$ of the closure of the locus of nonsingular L\"uroth
quartics.
