Consider the nine numbers from $1$ to $9$ inclusive: each digit appears once, with the exception of zero.

Now consider the $90$ two-digit numbers from $10$ to $99$ inclusive: each of the $10$ digits makes the same number of appearances as the second digit of a number and the digits from $1$ to $9$ make an equal number of appearances as the first digit of a number, but zero never appears as a first digit.

There is a similar pattern in the $900$ three-digit numbers from $100$ to $999$ inclusive with zero never appearing as a first digit, but making the same number of appearances as second or third digit as the other nine digits.

This leaves only the number $1000$ in which there are more zeros than any other digit, but not enough to make up for the fact that zero appears far fewer times than the other nine digits in the numbers less than $1000$. Moreover, the digit $1$ appears exactly once in $1000$ whereas the digits $2$ to $9$ do not appear.

Hence, the digit $1$ appears the largest number of times.

(You may wish to check that $0$ appears $192$ times, $1$ appears $301$ times and each of $2$ to $9$ appears $300$ times.)