Concept:
Hybridisation depends upon the steric number (number of \(\sigma\)-bonds and lone pairs) around the atom.
\[
\boxed{
\begin{aligned}
sp &\rightarrow 2 \text{ electron domains (linear)}\\
sp^2 &\rightarrow 3 \text{ electron domains (trigonal planar)}\\
sp^3 &\rightarrow 4 \text{ electron domains (tetrahedral)}
\end{aligned}
}
\]
The structure of acrylonitrile is
\[
CH_2=CH-C\equiv N.
\]
The carbon atom of the nitrile group forms a triple bond with nitrogen and is therefore \(sp\)-hybridised.
Step 1: Determine the hybridisation of the first carbon.
The first carbon is
\[
CH_2.
\]
It is involved in one double bond and two single bonds.
Hence, it has three electron domains.
Therefore,
\[
\boxed{C_1=sp^2.}
\]
Step 2: Determine the hybridisation of the second carbon.
The second carbon is attached by
• one double bond,
• one single bond to the nitrile carbon,
• one single bond to hydrogen.
Thus, it also has three electron domains.
Hence,
\[
\boxed{C_2=sp^2.}
\]
Step 3: Determine the hybridisation of the nitrile carbon.
The third carbon is
\[
C\equiv N.
\]
It forms
• one triple bond with nitrogen,
• one single bond with carbon.
Therefore, it possesses two electron domains.
Hence,
\[
\boxed{C_3=sp.}
\]
Thus, the hybridisation is
\[
\boxed{
C_1=sp^2,\;
C_2=sp^2,\;
C_3=sp.
}
\]
Therefore,
\[
\boxed{\textbf{Option (B)}}
\]
is the correct answer.