  This work focuses on higgledy-piggledy sets of $k$-subspaces in
$\text{PG}(N,q)$, i.e. sets of projective subspaces that are 'well-spread-out'.
More precisely, the set of intersection points of these $k$-subspaces with any
$(N-k)$-subspace $\kappa$ of $\text{PG}(N,q)$ spans $\kappa$ itself. We
highlight three methods to construct small higgledy-piggledy sets of
$k$-subspaces and discuss, for $k\in\{1,N-2\}$, 'optimal' sets that cover the
smallest possible number of points. Furthermore, we investigate small
non-trivial higgledy-piggledy sets in $\text{PG}(N,q)$, $N\leqslant5$. Our main
result is the existence of six lines of $\text{PG}(4,q)$ in higgledy-piggledy
arrangement, two of which intersect. Exploiting the construction methods
mentioned above, we also show the existence of six planes of $\text{PG}(4,q)$
in higgledy-piggledy arrangement, two of which maximally intersect, as well as
the existence of two higgledy-piggledy sets in $\text{PG}(5,q)$ consisting of
eight planes and seven solids, respectively. Finally, we translate these
geometrical results to a coding- and graph-theoretical context.
