Why is MnSO4 colourless?
Manganese(II) sulphate is not colourless. It is a faint pink. Compare this image taken from Wikipedia, where a full list of authors is available:



However, it is true that the colour is rather faint, and you need a white background to properly see it. That’s because manganese(II) is a $\mathrm{d^5}$ system and a high-spin one of those meaning that all five d-orbitals are occupied by a single electron:



Exciting an electron directly is forbidden due to the spin-retaining rule: The excited electron must occupy an already populated orbital, which is only possible if it flips its spin, which is forbidden by selection rules. Special mechanisms are needed to account for the (obviously possible but faint) transition here; the page Orthocresol linked in the comments names spin-orbit coupling.