Which test is used to distinguish between saturated and unsaturated hydrocarbons?

Bromine water has a red-brown colour because of the presence of bromine in it. When it is added to an unsaturated compound, then bromine gets added to the unsaturated compound and the red-brown colour of bromine is discharge. 

So, if an organic compound decolourises bromine water, then it will be an unsaturated compound while saturated ones do not decolourise it.

A glance at other options:

• The Lucas Test distinguish primary, secondary and tertiary alcohols.
• Tollens' test, also known as silver-mirror test, distinguish between an aldehyde and a ketone. 
• A chlorine test is generally given to determine the presence and amount of chlorine.