  Let $X_4\subset\mathbb{P}^{n+1}$ be a quartic hypersurface of dimension
$n\geq 4$ over an infinite field $k$. We show that if either $X_4$ contains a
linear subspace $\Lambda$ of dimension $h\geq \max\{2,\dim(\Lambda\cap
\text{Sing}(X_4))-2\}$ or has double points along a linear subspace of
dimension $h\geq 3$, a smooth $k$-rational point and is otherwise general, then
$X_4$ is unirational over $k$. This improves previous results by A. Predonzan
and J. Harris, B. Mazur, R. Pandharipande for quartics. We also provide a
density result for the $k$-rational points of quartic $3$-folds with a double
plane over a number field, and several unirationality results for quintic
hypersurfaces over a $C_r$ field.
