We report theoretical and numerical results on bifurcations in thermal instability for a viscoelastic fluid saturating a porous square cavity heated from below. Temporal stability analysis showed that the first bifurcation from the conductive state may be either oscillatory for sufficiently elastic fluids or stationary for weakly elastic fluids. The dynamics associated with the nonlinear interaction between the two kinds of instabilities is first analyzed in the framework of a weakly nonlinear theory. On the other hand, computations performed with high Rayleigh number for weakly and strongly elastic fluids indicated that the system exhibits successive bifurcations from stationary or oscillatory single-cell convection to a more complex spatio-temporal multi-cellular flows. The major new findings were presented in the form of bifurcations diagrams as functions of viscoelastic parameters for Rayleigh number up to 420.