We experimentally study the competition between Kerr beam self-cleaning and Raman beam cleanup in a multimode air-silica microstructure optical fiber. Kerr beam self-cleaning of the pump is observed for a certain range of input powers only. Stokes Raman beam generation and cleanup lead to both depletion and degradation of beam quality for the pump. The interplay of modal four-wave mixing and Raman scattering in the infrared domain leads to the generation of a multimode supercontinuum ranging from 500 nm up to 1800 nm.