Deep burnup and high minor actinides (MA) loading are two alluring features for molten salt reactors (MSR) to incinerate the nuclear wastes. The transmutation capability of minor actinides in MSR is tightly related with the neutron spectrum, the loading of MA and the carrier salt compositions. In this work, three MSR core designs (thermal, epithermal and fast) and two types of salt compositions (Flibe and Flinak) with different solubility limits of transuranic elements are chosen for analyzing the transmutation capability of MA. With a significant mole fraction of MA loading (4% in the Flibe salt and 10% in the Flinak salt) and the continuous MA refueling, MSR acquires an excellent transmutation rate. The specific incineration rates of MA in the thermal, epithermal and fast MSR cores with the Flibe salt are about 167, 185 and 206 kg/GWth/y, respectively. With a larger loading of MA in the Flinak salt, a higher annual incineration rate of MA can be obtained, which are about 170, 206 and 247 kg/GWth/y in thermal, epithermal and fast MSRs, respectively. On the other hand, since there is a preferred neutron economy for the Flibe salt, a higher MA incineration ratio is achieved than that for the Flinak salt. When the neutron spectrum varies from the thermal to fast region, the MA incineration ratio ranges from 0.79 to 0.82 for the Flibe salt and it ranges from 0.75 to 0.81 for the Flinak salt. The transmutation capability of MA in MSR is much higher than that in solid-fueled reactors (~20 kg/GW.y in a PWR), which can provide a feasible way for reducing the current nuclear wastes.