In this paper, new descriptor observer design approaches are presented for multivariable systems with measurement noises. If the system matrix is stable, the proposed modified proportional and derivative observer technique allows us not only to decouple the measurement noise in any forms completely, but also obtain accurate estimations of both system states and measurement noises. If the system is detectable and the noise is bounded, it has been shown that two design parameters can provide more degrees of freedom. In this context, the derivative gain parameter is chosen to reduce the amplification of the noise, and the proportional gain is then selected to guarantee the stability of the error dynamics. For systems with both measurement output noises and input disturbances, a modified proportional, derivative and integral observer is constructed to asymptotically estimate states, input disturbances, and output noises at the same time. The applications of the proposed observers for systems with actuator and sensor faults are also discussed. Numerical examples are given to illustrate the design procedures, and the simulation results show the satisfactory tracking performance.