Classical constrained systems can be obtained by symplectic reduction. Many of these, including Yang-Mills fields and gravity, are singular. The presence of singularities causes great difficulties in quantizing the systems, only because the quantized Hamiltonian is not essentially self-adjoint on its natural domain. A new approach, quantization via Rieffel induction, which is known to be the quantization of classical symplectic reduction, is used. This method is explicitly applied to many singular examples studied in the literature. In each case, this new approach correctly produces a well-defined, completely specified reduced Hamiltonian and reduced state space. We then study the reduction ofT*Gby the adjoint action ofG(takingG = SU(2),SU(3) as concrete examples). This comes from Yang-Mills theory on a circle. Again, the reduced (i.e. physical) quantum Hamiltonian and quantum state space are explicitly obtained. In particular, the reduced Hamiltonian is shown to be defined by Neumann boundary conditions.