We consider a normal lead coupled to a Majorana bound state. We show that the spin-resolved current correlations exhibit unique features which distinguish Majorana bound states from other low-energy resonances. In particular, the spin-up and spin-down currents from a Majorana bound state are anti-correlated at low bias voltages, and become uncorrelated at higher voltages. This behavior is independent of the exact form of coupling to the lead and of the direction of the spin polarization. In contrast, an ordinary low-energy Andreev bound state gives rise to a positive correlation between the spin-up and spin-down currents, and the spin-resolved current-current correlations approach a constant at high bias voltages. We discuss experimental setups in which this effect can be measured.