  Suppose $g_t$ is a $1$-parameter $\mathrm{Ad}$-diagonalizable subgroup of a
Lie group $G$ and $\Gamma < G$ is a lattice. We study the dimension of bounded
and divergent orbits of $g_t$ emanating from a class of curves lying on leaves
of the unstable foliation of $g_t$ on the homogeneous space $G/\Gamma$. We
obtain sharp upper bounds on the Hausdorff dimension of divergent on average
orbits and show that the set of bounded orbits is winning in the sense of
Schmidt (and, hence, has full dimension). The class of curves we study is
roughly characterized by being tangent to copies of $\mathrm{SL}(2,\mathbb{R})$
inside $G$, which are not contained in a proper parabolic subgroup of $G$.
  We describe applications of our results to problems in Diophantine
approximation by number fields and intrinsic Diophantine approximation on
spheres. Our methods also yield the following result for lines in the space of
square systems of linear forms: suppose $\varphi(s) = sY + Z$ where $Y\in
\mathrm{GL}(n,\mathbb{R})$ and $Z\in M_{n,n}(\mathbb{R})$. Then, the dimension
of the set of points $s$ such that $\varphi(s)$ is singular is at most $1/2$
while badly approximable points have Hausdorff dimension equal to $1$.
