We consider the problem of testing simultaneously many null hypotheses when the test statistics have a discrete distribution. We present new modifications of the Benjamini-Hochberg procedure that incorporate the discrete structure of the data in an appropriate way. These new procedures are theoretically proved to control the false discovery rate (FDR) for any fixed number of null hypotheses. A strong point of our FDR controlling methodology is that it allows to incorporate at once the discreteness and the quantity of signal of the data (so called "$\pi_0$-adaptation "). Finally, the power advantage of the new methods is demonstrated by using both numerical experiments and real data sets.