In this paper we find stability conditions of a two-class retrial system with unreliable server, in which the new customer joins a class-dependent orbit queue regardless of the state of the server. The interrupted customer joins the top of the corresponding orbit and tries to occupy the server after a class-dependent exponential retrial time. To find stability conditions, we apply two different approaches, regenerative approach and the stability analysis of a Markov chain (the MC approach) which has been developed in Fayolle et al. (Topics in the constructive theory of countable Markov chains, Cambridge University Press, Cambridge, 1995). The former approach allows to obtain a transparent and intuitive necessary stability condition. Then we apply the MC approach to obtain the stability criterion of the embedded two-dimensional Markov chain describing the state of orbits at the instances when the server becomes free. We use a necessary stability condition obtained by the regenerative method to present the stability criterion in a compact form. Moreover we discuss a modified controllable system and compare stability zones of these two systems. Some numerical examples based on simulation results are included as well which illustrate the theoretical issues.