During the beam commissioning of the Large Hadron Collider (LHC) [LHC Design Report No. CERN-2004-003-V-1, 2004 [http://cds.cern.ch/record/782076?ln=en]; O. Bruning, H. Burkhardt, and S. Myers, Prog. Part. Nucl. Phys. 67, 705 (2012)] with 150, 75, 50, and 25-ns bunch spacing, important electron-cloud effects, like pressure rise, cryogenic heat load, beam instabilities, or emittance growth, were observed. Methods have been developed to infer different key beam-pipe surface parameters by benchmarking simulations and pressure rise as well as heat-load observations. These methods allow us to monitor the scrubbing process, i.e., the reduction of the secondary emission yield as a function of time, in order to decide on the most appropriate strategies for machine operation. To better understand the influence of electron clouds on the beam dynamics, simulations have been carried out to examine both the coherent and the incoherent effects on the beam. In this paper we present the methodology and first results for the scrubbing monitoring process at the LHC. We also review simulated instability thresholds and tune footprints for beams of different emittance, interacting with an electron cloud in field-free or dipole regions.