We experimentally study the spatial beam profile and the spectral broadening at the output of a multimode air-silica microstructure fiber taper, used along the direction of an increasing fiber diameter. By using a laser pump at 1064 nm emitting 60 ps Gaussian beam pulses, we observed a competition between Raman beam cleanup and Kerr beam self-cleaning: the multimode frequency conversion process permits to generate spectral sidebands with frequency detuning from the pump that are difficult to obtain in standard graded-index multimode fibers. The generated supercontinuum spans from 500 nm up to 2.5 µm.