The list of one hundred numbers is this:

     1,  2, ...,  9, 10, 11, ..., 99, 100

We want to compare it to this list of 100 numbers:

00, 01, 02, ..., 09, 10, 11, ..., 99.

In the 2nd list there are 10 of each digit used as a first digit,
and 10 of each digit used as a second digit. So each digit is used
20 times in the second list.

Now we compare the second list to the first list:

For 10 through 99, the lists are the same.

There are 11 zeros in the second list among 00, 01,...09, that are
not used in the first list.  

So there are 20-11 or 9 0's up through 99 that are not used in the 
first list, plus there are two more zeros in 100.  

So there are only 11 zeros used in the first list, and 20 of 
all the other digits, except 1. 

There is one more 1 in the first list, because of the 1 in 100. 

So there are 21 1's, 11 0', and 20 of each of the other digits.

Answer: 0 is used least often (11 times). 

Edwin