By using the methods of the theory of two- and three-dimensional linear absolute and convective instabilities, we examine the nature of the instability at the onset of convection in a model of convection in an extended horizontal layer of a saturated porous medium with either horizontal or vertical salinity and inclined temperature gradients, and horizontal throughflow. First, normal modes are analysed and the critical values of the vertical thermal Rayleigh number,Rv, wavenumber vector, (k,l) and frequency, ω, are obtained for a variety of values of the horizontal thermal and salinity Rayleigh numbers,RhandSh, respectively, the vertical salinity Rayleigh numberSvand the horizontal Péclet number,Qh. In the computations, a high-precision pseudo-spectral Chebyshev-collocation method is used. In most of the cases of parameter combinations considered, the onset of convection occurs through a longitudinal mode. Most of the non-longitudinal critical modes are oscillatory. Further, it is revealed that there exists an absolute/convective instability dichotomy at the onset of three-dimensional convection in a set of the base states given by exact analytic solutions of the equations of motion in the model. This echoes the results of Brevdo (vol. 641, 2009, p. 475) for transverse modes in a model with inclined temperature gradient and vertical throughflow, but with no salinity. The dependence of the dichotomy on the inclined thermal gradient, and on the horizontal and the vertical salinity gradients is investigated, for the longitudinal modes treated both as two-dimensional as well as three-dimensional modes, and for the non-longitudinal modes. For a certain set of parameter cases, it was found that the destabilization through longitudinal modes treated as two-dimensional modes has the character of absolute instability whereas a three-dimensional analysis of these modes revealed that the instability is convective, with the group velocity vector of the emerging unstable wavepacket being parallel to the axis of the convection rolls. Since a similar effect was reported by Brevdo (vol. 641, 2009, p. 475) for a model with no salinity, we conclude that this effect is not a separate case. In most of the cases considered in which a marginally unstable base state is absolutely stable, but convectively unstable, the direction of propagation of the emerging unstable wavepacket is either parallel or perpendicular to the axis of the convection rolls. Only in the absolutely stable, but convectively unstable cases in which non-longitudinal modes are favourable, the angle, ϕ, between the group velocity vector of the unstable wavepacket and the axis of the rolls satisfies 0 < ϕ < 90°.