Step 1: Define vinyl polymerisation.
Vinyl polymerisation is chain-growth addition where monomers carrying a C=C double bond open up and link together with no small molecule lost.
Step 2: Set the test.
A monomer can do vinyl polymerisation only if it has a carbon-carbon double bond.
Step 3: Check vinyl cyanide.
Acrylonitrile $\text{CH}_2\text{=CH-CN}$ has a C=C, so it adds up into polyacrylonitrile.
Step 4: Check tetrafluoroethylene.
$\text{CF}_2\text{=CF}_2$ has a C=C and polymerises to Teflon.
Step 5: Check ethylene.
$\text{CH}_2\text{=CH}_2$ has a C=C and gives polyethylene.
Step 6: Check caprolactam.
$\varepsilon$-caprolactam is a cyclic amide with no C=C; it polymerises by ring opening to Nylon-6, not by the vinyl route.
Step 7: Pick the exception.
Only $\varepsilon$-caprolactam lacks the vinyl double bond, so it cannot undergo vinyl polymerisation, the answer being option (1).
\[ \boxed{\varepsilon\text{-Caprolactam (no C=C, ring-opening only)}} \]