At this time there are no direct tests of length contraction, as measuring the length of a moving object to the precision required has not been feasible. There is, however, a demonstration that it occurs:
A current-carrying wire is observed to be electrically neutral in its rest frame, and a nearby charged particle at rest in that frame is unaffected by the current. A nearby charged particle that is moving parallel to the wire, however, is subject to a magnetic force that is related to its speed relative to the wire. If one considers the situation in the rest frame of a charge moving with the drift velocity of the electrons in the wire, the force is purely electrostatic due to the different length contractions of the positive and negative charges in the wire (the former are fixed relative to the wire, while the latter are mobile with drift velocities of a few mm per second). This approach gives the correct quantitative value of the magnetic force in the wire frame. This is discussed in more detail in: Purcel, Electricity and Magnetism. It is rather remarkable that relativistic effects for such a tiny velocity explain the enormous magnetic forces we observe.