Step 1: Understanding the Concept:
Transition elements exhibit variable oxidation states because they can lose electrons from both their outermost $s$-orbital and their penultimate $d$-orbital. The number of oxidation states depends on the number of unpaired electrons in the $d$-subshell.
Step 2: Detailed Explanation:
1. Scandium (Sc): Has electronic configuration $[Ar] 3d^1 4s^2$. It typically shows only the $+3$ state.
2. Iron (Fe): Shows $+2, +3, +4, +6$ states (common are $+2$ and $+3$).
3. Zinc (Zn): Has a completely filled $d$-orbital ($3d^{10}$). It only shows the $+2$ state.
4. Manganese (Mn): Has electronic configuration $[Ar] 3d^5 4s^2$. With 5 unpaired $d$-electrons and 2 $s$-electrons, it can show the widest range of oxidation states from $+2$ to $+7$.
Step 3: Final Answer
Manganese shows the highest number of oxidation states.