Suppose there is a mass attached to a spring which is hanging from a fixed ceiling. The spring will be elongated and this system has potential energy stored because of the elongation. Now if the spring is cut at the topmost point what happens to the elastic potential energy stored in the spring-mass system?
The center of mass of the spring+mass system will accelerate downwards at $g$. During this motion the spring will contract, so the top end of the spring will accelerate faster than $g$, and initially the bottom end will hardly accelerate at all because at the initial moment the spring tension and the weight exactly cancel. There are some nice videos on youtube which show this effect (https://www.youtube.com/watch?v=uiyMuHuCFo4&frags=pl%2Cwn for example) - the lower end of the spring seems to "defy gravity" for a while. So as usual the potential energy of the spring is being converted into kinetic energy. When the spring is fully contracted it will expand again, performing simple harmonic motion in the rest frame of the center of mass. It will take a very long drop to actually see this oscillation in practice though.