The combination of strong magnetic fields and fast rotation is often invoked as a characteristic of the central engine for outstanding sources such as GRBs, hypernovae, and superluminous supernovae. However, the actual properties of the magnetic field during the collapse of the stellar progenitor are very uncertain, since they depend on the evolution of the star and can be affected by complex dynamo processes occurring in the central proto-neutron star. Using 3D relativistic MHD models we show that higher-order multipolar fields can lead to the onset of a supernova, although they tend to produce less energetic explosions and less collimated outflows. Quadrupolar fields efficiently extract angular momentum from the central core, but the rotational energy is partly stored in the equatorial regions, rather than powering up the polar outflows. Finally, our results show a strong magnetic quenching of the hydrodynamic non-axisymmetric instabilities that are associated to the emission of GWs.