Step 1: Understanding the Concept:
In qualitative inorganic analysis, metal cations are precipitated as their sulfides to separate and identify them.
Different metal sulfides exhibit characteristic colors that help in their identification.
Step 2: Key Formula or Approach:
Identify the standard colors of the precipitates of the given metal sulfides:
Lead sulfide ($PbS$), Nickel sulfide ($NiS$), and Cobalt sulfide ($CoS$) are well-known black precipitates.
Manganese sulfide ($MnS$) has a distinctly different color.
Step 3: Detailed Explanation:
Let us look at the characteristic colors of the options provided:
Lead sulfide ($PbS$) $\rightarrow$ Black precipitate.
Nickel sulfide ($NiS$) $\rightarrow$ Black precipitate.
Cobalt sulfide ($CoS$) $\rightarrow$ Black precipitate.
Manganese sulfide ($MnS$) $\rightarrow$ Flesh-colored or buff/pink precipitate.
Therefore, among the given options, $MnS$ is the sulfide that is not black.
Step 4: Final Answer:
The sulfide that is not black is MnS.