Q.Which of the following oxidation states is most common among the lanthanoids ?

Because the 4f orbitals are shielded by the filled $5s^25p^6$ subshells and contribute little to bonding, lanthanoids characteristically lose the two $6s$ electrons and one $4f/5d$ electron to give the common, most stable $+3$ oxidation state.

Although a few lanthanoids also show $+2$ (e.g. $Eu^{2+}$, $Yb^{2+}$, where a half-filled or fully-filled 4f subshell gives extra stability) or $+4$ (e.g. $Ce^{4+}$, $Tb^{4+}$, similarly stabilised by an empty or half-filled 4f configuration), these are exceptions restricted to specific elements. Across the whole lanthanoid series, $+3$ is the state shown by every member and is by far the most common and characteristic oxidation st …