Let $F$ be a nonquasi-unipotent one-parameter (cyclic) subgroup of a unimodular Lie group $G$, $\Gamma$ a discrete subgroup of $G$. We prove that for certain classes of subsets $Z$ of the homogeneous space $G/\Gamma$, the set of points in $G/\Gamma$ with $F$-orbits staying away from $Z$ has full Hausdorff dimension. From this we derive applications to geodesic flows on manifolds of constant negative curvature.