  We study certain physically-relevant subgeometries of binary symplectic polar
spaces $W(2N-1,2)$ of small rank $N$, when the points of these spaces
canonically encode $N$-qubit observables. Key characteristics of a subspace of
such a space $W(2N-1,2)$ are: the number of its negative lines, the
distribution of types of observables, the character of the geometric hyperplane
the subspace shares with the distinguished (non-singular) quadric of
$W(2N-1,2)$ and the structure of its Veldkamp space. In particular, we classify
and count polar subspaces of $W(2N-1,2)$ whose rank is $N-1$. $W(3,2)$ features
three negative lines of the same type and its $W(1,2)$'s are of five different
types. $W(5,2)$ is endowed with 90 negative lines of two types and its
$W(3,2)$'s split into 13 types. 279 out of 480 $W(3,2)$'s with three negative
lines are composite, i.\,e. they all originate from the two-qubit $W(3,2)$.
Given a three-qubit $W(3,2)$ and any of its geometric hyperplanes, there are
three other $W(3,2)$'s possessing the same hyperplane. The same holds if a
geometric hyperplane is replaced by a `planar' tricentric triad. A hyperbolic
quadric of $W(5,2)$ is found to host particular sets of seven $W(3,2)$'s, each
of them being uniquely tied to a Conwell heptad with respect to the quadric.
There is also a particular type of $W(3,2)$'s, a representative of which
features a point each line through which is negative. Finally, $W(7,2)$ is
found to possess 1908 negative lines of five types and its $W(5,2)$'s fall into
as many as 29 types. 1524 out of 1560 $W(5,2)$'s with 90 negative lines
originate from the three-qubit $W(5,2)$. Remarkably, the difference in the
number of negative lines for any two distinct types of four-qubit $W(5,2)$'s is
a multiple of four.
