Step 1: Analyze the given coordination compound [Co(NH$_3$)$_5$(NO$_2$)] (NO$_3$)$_2$.
The complex ion is [Co(NH$_3$)$_5$(NO$_2$)]$^{+2}$ and the counter ions are 2 NO$_3^-$. The central metal ion is Co, with ligands 5 NH$_3$ and one NO$_2^-$.
Step 2: Check for each type of isomerism.
i. Optical Isomerism: Optical isomerism occurs in chiral complexes, which are non-superimposable mirror images of each other. Octahedral complexes with at least two different types of ligands can exhibit optical isomerism if they lack a plane of symmetry. The given complex has five identical NH$_3$ ligands and one NO$_2^-$ ligand, so it is unlikely to be chiral. To be certain, consider the possible arrangements. There are no non-superimposable mirror images possible for this configuration. Thus, optical isomerism is not exhibited.
ii. Linkage Isomerism: Linkage isomerism occurs when an ambidentate ligand (a ligand that can bind to the metal ion through two different atoms) is present. The nitrite ion (NO$_2^-$) is an ambidentate ligand; it can bind through the nitrogen atom (-NO$_2$, nitro) or through one of the oxygen atoms (-ONO, nitrito). Therefore, this complex can exhibit linkage isomerism.
iii. Ionization Isomerism: Ionization isomerism occurs when there is an exchange of ligands between the coordination sphere and the counter ion sphere. In this complex, NO$_2^-$ is a ligand in the coordination sphere, and NO$_3^-$ is the counter ion. An ionization isomer would involve NO$_3^-$ as a ligand and NO$_2^-$ as a counter ion, such as [Co(NH$_3$)$_5$(NO$_3$)] (NO$_2$)(NO$_3$). This is possible, so ionization isomerism is exhibited.
iv. Coordination Isomerism: Coordination isomerism occurs in salts where both the cation and anion are complex ions, and there is an exchange of ligands between the two complex ions. The given compound has a complex cation and a simple anion (nitrate), so coordination isomerism is not possible.
\end{itemize}
Step 3: Identify the types of isomerism exhibited.
The complex exhibits linkage isomerism (due to NO$_2^-$) and ionization isomerism (exchange of NO$_2^-$ and NO$_3^-$).
Final Answer:
\[
\boxed{\text{ii, iii only}}
\]