In this paper, the new design problems of full and reduced-order $H_{infty}$ filters including static filters are investigated for discrete-time singular systems with lossy measurements. The lossy measurement is described by a stochastic variable satisfying Bernoulli random binary distribution to model the measured output. Being different from the previous results, the filter existence condition by a new Lyapunov function for singular systems with lossy measurements is given using a strict linear matrix inequality (LMI) in terms of all variables. The proposed full and reduced-order $H_{infty}$ filter guarantees regularity, causality, and mean-square stochastic stability with a $H_{infty}$ performance bound of filtering error singular system. Moreover, it is shown that the reduced-order $H_{infty}$ filter design method is general for both singular systems and non-singular systems with lossy measurements. Numerical examples are worked out to illustrate the effectiveness and applicability of the proposed method.