Friendly measures, homogeneous flows and singular vectors

We prove that singular vectors have measure zero with respect to any friendly measure on ${\mathbb{R}}^n$ (e.g. the volume measure on a nondegenerate submanifold). This generalizes special cases considered by Davenport-Schmidt, Baker and Bugeaud. The main tool is quantitative nondivergence estimates for quasi-polynomial flows on homogeneous spaces.