The effects of 2D tunneling bifurcations for quasi-one-dimensional and quasi-two-dimensional Au-quantum dot (QD) arrays in thin dielectric films in an external electric field have been studied theoretically and experimentally by Conductive Atomic Force Microscopy (CAFM). In the case of quasi-one-dimensional Au-QD structures (with the QD size similar to 5 nm), in a dielectric film, a single break under positive bias polarity, corresponding to the effect of 2D tunneling bifurcation, previously predicted theoretically by our team , has been detected in experimental I-V curves of the CAFM probe-to-sample contact. A convincing qualitative agreement between the obtained experimental I-V curves and the theoretical field dependence for the 2D-dissipative tunneling probability in the model 2D-oscillator potential has been obtained for the case of parallel tunneling in the weak-dissipation limit at a finite temperature in an external electric field. In the case of quasi-two-dimensional structures with Au-QD (with the QD sizes of 2 to 5 nm), possessing metamaterial properties, a pair of kinks corresponding to the double effect of 2D-tunneling bifurcations has been detected on the experimental I-V curves. A qualitative agreement between the experimental I-V curves and the theoretical field dependence for the 2D-dissipative tunneling probability has been obtained for a situation with an effectively "negative" permittivity of the heat bath.