The Kelvin-Helmholtz model for the evolution of an infinitesimally thin vortex sheet in an inviscid fluid is mathematically ill-posed for general classes of initial conditions. However, if the initial data, say imposed at t=0, is in a certain class of analytic functions then the problem is well-posed for a finite time until a singularity forms, say at t=ts, on the vortex-sheet interface, e.g. as illustrated by Moore (1979). However, if the problem is analytically continued into the complex plane, then the singularity, or singularities, exist for t<ts away from the physical real axis. More specifically, Cowley et al. (1999) found that for a class of analytic initial conditions, singularities can form in the complex plane at t=0+. They posed asymptotic expansions in the neighbourhood of these singularities for 0<t<<1, and found numerical solutions to the governing similarity differential equations. In this paper we obtain new exact solutions to these equations, show that the singularities always correspond to local 3/2-power singularities, and determine both the number and precise locations of all branch points. Further, our analytical approach can be extended to a more general class of initial conditions. These new exact solutions can assist in resolving the small-time behaviour for the numerical solution of the Birkhoff-Rott equations.