  We prove inheritance of measure zero property of the set of singular vectors
for affine subspaces and submanifolds inside those affine subspaces. We define
a notion of $n$-singularity for matrices, which is closely related to the
uniform exponent of irrationality. For certain affine subspaces, we show that
the set of singular vectors has measure zero if and only if the parametrizing
matrix is not $n$-singular. In particular, we show for affine hyperplanes the
set of singular vectors has measure zero if and only if the parametrizing
matrix is not rational.
