The only way that a distribution has a standard deviation equal to exactly 0 is that none that is different. This is, all the data has the same value.
That is because the standard deviation is the square root of the variance and the variance is a sum of squares terms. If just one term of the variance is different from 0 the result will never be exactly 0. So, you can conclude that if a distribuition has a standard deviation equal to exactly o is because all the data are exactly equal.