Three-dimensional (3D) particle localization at the nanometer scale plays a central role in 3D particle tracking and 3D localization-based super-resolution microscopy. Here we introduce a localization algorithm that is independent of theoretical models and therefore generally applicable to a large number of experimental realizations. Applying this algorithm and a convertible experimental setup we compare the performance of the two major 3D techniques based on astigmatic distortions and on multiplane detection. In both methods we obtain experimental 3D localization accuracies in agreement with theoretical predictions and characterize the depth dependence of the localization accuracy in detail.