The longitudinal motion of electron in a storage ring serves as an interesting example of a one-dimensional mechanical system. This system had been studied well many years ago and described in any accelerator textbook. On the other hand, this system is one of the simplest, like the hydrogen atom, or an electron in Paul and Penning electromagnetic traps. Detecting synchrotron radiation of a single particle enables a number of interesting studies on the particle's longitudinal state and motion in a giantic trap of the storage ring. I will discuss experiments with a single electron carried out at VEPP-3 storage ring at Budker Institute of Nuclear Physics in 1991-1993. In the first experiment we have measured the localization length of a single electron captured in the longitudinal potential well. In the second experiment we studied the stochastic process of the electron's synchrotron oscillations driven by random acts of the quanta emissions and radiation damping. In conclusion I will discuss some considerations for design of a storage ring, optimized for experiments with a single electron.