It is a great challenge to detect singular cycles and chaos in dynamical systems with multiple discontinuous boundaries. This paper takes the challenge to investigate the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles connecting saddle-focus equilibriums, in a new class of three-dimensional three-zone piecewise affine systems. It develops a method to accurately predict the coexisting homoclinic and heteroclinic cycles in such a system. Furthermore, this paper establishes some conditions for chaos to exist in the system, with rigorous mathematical proof of chaos emerged from the coexistence of these singular cycles. Finally, it presents numerical simulations to verify the theoretical results.