The local stability of shear flow of nematic liquid crystals with a positive Leslie viscosity alpha(3) is investigated. Using abstract techniques we show, that there exists a physically realizable in-plane flow for low enough shearing rates. Lower bounds for the onset of instabilities as well as approximate expressions for their thresholds are derived analytically. Both the tumbling instability and the stability of the director against fluctuations which tend to bring it out of the shearing plane are discussed. Numerical solutions to the in-plane problem are presented for different combinations of the crucial parameter xi = alpha(3)/\alpha(2)\ for both perpendicular and parallel boundary conditions. Some general features of these solutions are exhibited. By comparing with the previously derived analytical conditions for tumbling it is proven that the theory works well within its limitations. Regarding the out of plane instability, no definite answers are given but merely a lower bound for its onset is derived. It is shown that this lower bound always exceeds the threshold for the onset of tumbling, thus guaranteeing that tumbling really occurs. The calculations are performed taking transverse flow effects into account. However, it is shown that these do only affect the results that are derived when they are neglected in a minor way. (C) 2004 Elsevier B.V. All rights reserved.