Given an alternating trilinear form $${T\in {\rm Alt}(\times^{3}V_{6})}$$ on V 6 = V(6, 2) let $${\mathcal{L}_{T}}$$ denote the set of those lines $${\langle a, b \rangle}$$ in $${{\rm PG}(5,2)=\mathbb{P}V_{6}}$$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all $${x\in {\rm PG}(5, 2).}$$ If $${\mathcal{L}_{21}}$$ is a Desarguesian line-spread in PG(5, 2) it is shown that $${\mathcal{L}_{T}=\mathcal{L}_{21}}$$ for precisely three choices T 1,T 2,T 3 of T, which moreover satisfy T 1 + T 2 + T 3 = 0. For $${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$$ the $${\mathcal{G}_{T}}$$ -orbits of flats in PG(5, 2) are determined, where $${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each $${\mathcal{G}_{T}}$$ -orbit, the T-associate U # is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .