Step 1: State what each listed test is actually for.
Fehling's test and Tollens' test both detect aldehydes, not alcohols. Hinsberg's test is used on amines, not alcohols. That leaves the Lucas test as the only one built for alcohols.
Step 2: Recall what the Lucas reagent is and does.
The Lucas reagent is a mix of concentrated $\text{HCl}$ and anhydrous $\text{ZnCl}_2$. It converts an alcohol's $-\text{OH}$ into $-\text{Cl}$, and the alkyl chloride formed is insoluble in water, so it shows up as cloudiness.
Step 3: Link the speed of cloudiness to carbocation stability.
A tertiary alcohol forms the most stable carbocation, so it reacts and turns cloudy at once. A secondary alcohol takes about 5 to 10 minutes. A primary alcohol barely reacts at room temperature and needs heating to show any turbidity.
Step 4: State the conclusion.
Because the three classes give three clearly different reaction times with the same reagent, the Lucas test is the correct way to distinguish them.
\[ \boxed{\text{Lucas test}} \]