The smallest jet engine measures just 600nm across and weighs 1 femtogram (10^-15 kg). It was produced by Leibniz Institute Solid State Physics Dresden and demonstrated in Dresden, Germany, in September 2010.
The 'engine' is actually a conical nanotube with an inner layer of platinum. When placed in a solution of hydrogen peroxide the platinum catalyses the solution producing thrust that pushes the tiny engine along. While it's not a method of propulsion that works in the real world, nonetheless it technically qualifies as a jet engine because of the way it produces thrust.
Comments below may relate to previous holders of this record.