Step 1: Define linkage isomerism.
Linkage isomerism occurs when a ligand can bind to the metal center in more than one way. This typically happens with ligands that have more than one donor atom, such as NO$_2^-$, SCN$^-$, and CN$^-$.
Step 2: Analyze the options.
- NO$_2^-$ can bind through the nitrogen atom or the oxygen atom, so it can exhibit linkage isomerism.
- SCN$^-$ can bind through either the sulfur or nitrogen atom, so it can also show linkage isomerism.
- CN$^-$ can bind through the carbon atom or the nitrogen atom, so it can show linkage isomerism.
- NH$_3$, however, only binds through the nitrogen atom and does not show linkage isomerism.
Step 3: Apply to the options.
Since NH$_3$ cannot show linkage isomerism, the correct answer is (D).
Final Answer:
\[
\boxed{\text{NH}_3}
\]