Some noteworthy alternating trilinear forms

Given an alternating trilinear form T on the V=F^n let L_T denote the set of T-singular lines in the projective space PV. These are all lines <a,b> for which the linear form T(a,b,.) is identically zero. Amongst the immense profusion of different kinds of T we single out a few which we deem noteworthy by virtue of the special nature of their set LT.