Since the discovery of the Higgs boson in 2012, the precise determination of its properties and interaction with other particles remains a central focus of research at the CERN LHC. Differential cross section measurements of variables characterizing the Higgs boson across a number of its decay modes are performed, scrutinizing the theoretical modeling across a broad range of kinematic spectra as well as probing for the presence of physics beyond the standard model. Sophisticated observables are developed to probe the potential presence of new physics at energy scales higher than those directly accessible at the LHC, modifying the Higgs boson coupling to other fundamental particles. A number of results, public as of June 2024, reporting differential measurements and searches for the effect of effective field theory operators using 138 fb$^{-1}$ of data collected by the CMS experiment in proton-proton collisions at a center-of-mass energy of 13 TeV is presented.