What is the largest dimensional linear space of singular planar cubics? Is this known?

Think of the space of planar cubics as a PP^9 (parametrized by the coefficients). The discriminant \Delta is then a degree 12 polynomial on PP^9 whose vanishing parametrizes singular cubic curves. What is the dimension of the largest linear space inside the vanishing of \Delta? I attempted doing this on a computer, but the computations were too large to terminate.