An approximate Schwinger-Dyson equation for the quark propagator in Euclidean QCD is solved numerically in the complex s = p2 plane, where p(mu) is the quark Euclidean four-momentum. Complex conjugate pairs of singularities are discovered and an analytic contour pinch analysis shows that any such singularities must be logarithmic branch points. Appropriate logarithmic functions are fitted near the branch points to accurately determine their position. The physical significance of these singularities is as yet unclear.