We demonstrate how spatial beam self-cleaning and supercontinuum generation in graded-index multimode optical fibers can be directly applied in multiplex coherent anti-Stokes Raman Scattering (M-CARS) spectroscopy. Although supercontinuum generation causes pump depletion mainly in the center of the beam, the partial recovery of the pump brightness due to self-cleaning may enable self-referenced M-CARS, with no additional delay lines to synchronize pump and Stokes waves. As a proof-of-principle, we report examples of imaging of single chemical compounds and polystyrene beads. The new scheme paves the way towards simpler M-CARS systems based on multimode fiber sources.