Step 1: Recall what mustard gas is.
Mustard gas is the chemical warfare agent sulfur mustard, named bis(2-chloroethyl) sulfide, an organosulfur thioether bearing two chloroethyl arms.
Step 2: Write its formula.
Its structure is $\mathrm{S(CH_2CH_2Cl)_2}$, a sulfur atom linking two $\mathrm{-CH_2CH_2Cl}$ groups.
Step 3: Identify option A.
$\mathrm{SOCl_2}$ is thionyl chloride, a lab reagent for making alkyl chlorides, not a war gas.
Step 4: Identify option B.
$\mathrm{COCl_2}$ is phosgene, a carbonyl chloride; toxic, but distinct from mustard gas.
Step 5: Identify option D.
$\mathrm{CCl_3NO_2}$ is chloropicrin, used as a soil fumigant and tear agent.
Step 6: Conclude.
The formula of mustard gas is $\mathrm{S(CH_2CH_2Cl)_2}$, option (C).
\[ \boxed{\mathrm{S(CH_2CH_2Cl)_2}} \]