We classify spreads of the Tits quadrangles
${T}_{2}\left(\mathcal{O}\right)$, for
$\mathcal{O}$ an oval
in
$\mathsf{PG}\left(2,q\right)$, for
$q=2,4,8,16$ and
$32$,
using a computer for the last three cases. Along the way, we classify
$\alpha $flocks of
$\mathsf{PG}\left(3,32\right)$, and so flocks of the
quadratic cone in
$\mathsf{PG}\left(3,32\right)$.
Perhaps our most striking results are that, for many ovals
$\mathcal{O}$ in
$\mathsf{PG}\left(2,32\right)$, including all 12
O’KeefePenttila ovals,
${T}_{2}\left(\mathcal{O}\right)$ has no
spreads, and that
${T}_{2}\left(\mathcal{O}\right)$ is a proper
subGQ of a GQ of order
$\left(s,32\right)$ for
precisely 6 of the 35 ovals
$\mathcal{O}$
of
$\mathsf{PG}\left(2,32\right)$, all of
which were previously known to be subquadrangles of a (flock or dual Tits) GQ of order
$\left(1024,32\right)$. Also
${T}_{2}\left(\mathcal{O}\right)$ is not a proper subGQ
of a GQ of order
$\left(s,q\right)$ or
of a GQ of order
$\left(q,t\right)$
for
$\mathcal{O}$ a pointed
conic in
$\mathsf{PG}\left(2,q\right)$,
for
$q=16,32$.
