For the last two decades, experimental information on nuclear level densities for about 60 different nuclei has been obtained on the basis of the Oslo method. While each of these measurements has been typically compared to one or a few level density models, a global study including all the measurements has been missing. The present study provides a systematic comparison between Oslo data and six global level density models for 42 nuclei for which $s$-wave resonance spacings are also available. We apply a coherent normalization procedure to the Oslo data for each of the six different models, all being treated on the same footing. Our quantitative analysis shows that the constant-temperature model presents the best global description of the Oslo data, closely followed by the mean-field plus combinatorial model and Hartree-Fock plus statistical model. Their accuracies are quite similar, so that it remains difficult to clearly favour one of these models. When considering energies above the threshold where the experimental level scheme is complete, all the six models are shown to lead to rather similar accuracies with respect to Oslo data. The recently proposed shape method can, in principle, improve the situation since it provides an absolute estimate of the excitation-energy dependence of the measured level densities. We show for the specific case of $^{112}$Cd that the shape method could exclude the Hartree-Fock plus statistical model. Such an analysis remains to be performed for the bulk of data for which the shape method can be applied to the Oslo measurements before drawing conclusions on the general quality of a given nuclear level density model.

11 pages, 7 figures, published in PRC