How many zeros ?

For the one-digit numbers including $0,$ There is only one $0,$ and that is $0$ itself.

For the two-digit numbers, only numbers ending in $0$ are valid; therefore there are nine $0$ s.

For the three-digit numbers, there are two $0$ s at every multiple of $100,$ one $0$ at other multiples of $10,$ and nine other $0$ s in the middle digit of the first ten numbers of every hundred. This adds up to twenty $0$ s per hundred, or $180$ $0$ s total.

There are three $0$ s in $1000.$

So the total number of $0$ s is $1 + 9 + 180 + 3 = \boxed{193}$