1. We check the hybridization of carbon atoms in each compound.

2. Option A: Acetone

   Structure: $\mathrm{CH_3-CO-CH_3}$

   The carbonyl carbon $\mathrm{C=O}$ is $sp^2$ hybridized.

   So, acetone has an $sp^2$ hybridized carbon.

3. Option B: Acetamide

   Structure: $\mathrm{CH_3-CONH_2}$

   The carbonyl carbon in the amide group is $sp^2$ hybridized.

   So, acetamide has an $sp^2$ hybridized carbon.

4. Option C: Acetonitrile

   Structure: $\mathrm{CH_3-C\equiv N}$

   The nitrile carbon (the carbon in $\mathrm{C\equiv N}$) is $sp$ hybridized, not $sp^2$. The methyl carbon is $sp^3$ hybridized.

   Therefore, acetonitrile does not have any $sp^2$ hybridized carbon.

5. Option D: Acetic acid

   Structure: $\mathrm{CH_3-COOH}$

   The carboxyl carbon is $sp^2$ hybridized.

   So, acetic acid has an $sp^2$ hybridized carbon.

6. Hence, the compound which does not have an $sp^2$ hybridized carbon is:

$\boxed{\text{C: Acetonitrile}}$