Multiplicity fluctuations of positively, negatively, and all charged hadrons in the forward hemisphere were studied in central Pb+Pb collisions at 20A,30A,40A,80A, and 158A GeV. The multiplicity distributions and their scaled variances ω are presented as functions of their dependence on collision energy as well as on rapidity and transverse momentum. The distributions have bell-like shapes and their scaled variances are in the range from 0.8 to 1.2 without any significant structure in their energy dependence. No indication of the critical point in fluctuations are observed. The string-hadronic ultrarelativistic quantum molecular dynamics (UrQMD) model significantly overpredicts the mean, but it approximately reproduces the scaled variance of the multiplicity distributions. The predictions of the statistical hadron-resonance gas model obtained within the grand-canonical and canonical ensembles disagree with the measured scaled variances. The narrower than Poissonian multiplicity fluctuations measured in numerous cases may be explained by the impact of conservation laws on fluctuations in relativistic systems.