  We study the Folklore set of Dirichlet improvable matrices in $\mathbb
R^{m\times n}$ which are neither singular nor badly approximable. We prove the
non-emptiness for all positive integer pairs $m,n$ apart from $\{m,n\}=\{
1,1\}$ and $\{m,n\}=\{ 2,3\}$ in a constructive way. For a wide range of
integer pairs $(m,n)$ we construct subsets of the Folklore set with an exact
prescribed Dirichlet constant (in some right neighbourhood of $0$). This
enables us to provide information on the Dirichlet Spectrum of matrices. The
key technique of our construction is to build certain $2$-dimensional
simultaneously approximable very singular vectors, and then `lift' these
vectors to higher dimensions by `glueing' them to a large set of well-behaved
matrices. Our technique is also applicable to arbitrary norms. As a corollary,
we obtain lower bounds on the Hausdorff dimension of these sets. These
statements complement previous results of the middle-named author (Selecta
Math. 2023), Beresnevich et. al. (Adv. Math. 2023), and Das et. al. (Adv. Math.
2024).
