The probabilities for gaps in the eigenvalue spectrum of finite N×N random unitary ensembles on the unit circle with a singular weight, and the related Hermitian ensembles on the line with Cauchy weight, are found exactly. The finite cases for exclusion from single and double intervals are given in terms of second-order second-degree ordinary differential equations (ODEs) which are related to certain Painlevé-VI transcendents. The scaled cases in the thermodynamic limit are again second degree and second order, this time related to Painlevé-V transcendents. Using transformations relating the second-degree ODE and transcendent we prove an identity for the scaled bulk limit which leads to a simple expression for the spacing probability density function. We also relate all the variables appearing in the Fredholm determinant formalism to particular Painlevé transcendents, in a simple and transparent way, and exhibit their scaling behaviour.