Step 1: Recall the carbylamine test.
The carbylamine test uses chloroform and alcoholic KOH. It gives isocyanides, which have a very bad, offensive smell.
Step 2: Know which amines respond.
Only primary amines ($R-NH_2$), which have an $-NH_2$ group with two hydrogens on nitrogen, give a positive carbylamine test. Secondary and tertiary amines do not.
Step 3: Check the tertiary amine.
N,N-Dimethylaniline is a tertiary amine, so it has no $N-H$ hydrogens to react. It gives a negative test.
Step 4: Check the secondary amine.
N-Methyl-o-methylaniline is a secondary amine, so it also fails the test and is negative.
Step 5: Find the primary amines.
2,4-Dimethylaniline and p-Methylbenzylamine both carry a free $-NH_2$ group, so they are primary amines and give a positive carbylamine test with the offensive odour. These match II and IV.
Step 6: State the answer.
So the compounds that respond are II and IV.
\[ \boxed{\text{II, IV only}} \]