  We investigate some aspects of the geometry of two classical generalisations
of the Hilbert schemes of points. Precisely, we show that parity conjecture for
$\text{Quot}_r^d\mathbb{A}^3$ already fails for $d=8$ and $r=2$ and that lots
of the elementary components of the nested Hilbert schemes of points on smooth
quasi-projective varieties of dimension at least 4 are generically non-reduced.
We also deduce that nested Hilbert schemes of points on smooth surfaces have
generically non-reduced components. Finally, we give an infinite family of
elementary components of the classical Hilbert schemes of points.
