  Let $L$ and $M$ denote the classical Lagrange and Markov spectra,
respectively. It is known that $L\subset M$ and that $M\setminus
L\neq\varnothing$. Inspired by three questions asked by the third author in
previous work investigating the fractal geometric properties of the Lagrange
and Markov spectra, we investigate the function $d_{loc}(t)$ that gives the
local Hausdorff dimension at a point $t$ of $L'$. Specifically, we construct
several intervals (having non-trivial intersection with $L'$) on which
$d_{loc}$ is non-decreasing. We also prove that the respective intersections of
$M'$ and $M''$ with these intervals coincide. Furthermore, we completely
characterize the local dimension of both spectra when restricted to those
intervals. Finally, we demonstrate the largest known elements of the difference
set $M\setminus L$ and describe two new maximal gaps of $M$ nearby.
