The divergence of the heat conductivity in the thermodynamic limit is investigated in 2d-lattice models of anharmonic solids with nearest-neighbour interaction from single-well potentials. Two different numerical approaches based on nonequilibrium and equilibrium simulations provide consistent indications in favour of a logarithmic divergence in ``ergodic'', i.e., highly chaotic, dynamical regimes. Analytical estimates obtained in the framework of linear-response theory confirm this finding, while tracing back the physical origin of this anomalous transport to the slow diffusion of the energy of hydrodynamic modes. Finally, numerical evidence of superanomalous transport is given in the weakly chaotic regime, typically observed below a threshold value of the energy density.