  According to Dyson's three fold way, from the viewpoint of global time
reversal symmetry there are three circular ensembles of unitary random matrices
relevant to the study of chaotic spectra in quantum mechanics. These are the
circular orthogonal, unitary and symplectic ensembles, denoted COE, CUE and CSE
respectively. For each of these three ensembles and their thinned versions,
whereby each eigenvalue is deleted independently with probability $1-\xi$, we
take up the problem of calculating the first two terms in the scaled large $N$
expansion of the spacing distributions. It is well known that the leading term
admits a characterisation in terms of both Fredholm determinants and Painlev\'e
transcendents. We show that modifications of these characterisations also
remain valid for the next to leading term, and that they provide schemes for
high precision numerical computations. In the case of the CUE there is an
application to the analysis of Odlyzko's data set for the Riemann zeros, and in
that case some further statistics are similarly analysed.
