You can ask lots of questions about primes. After posting 50 facts about primes, I couldn't resist making another graph. In this one, the x-axis is $n$ and the y-axis is the number of primes up to $n$ that contain a given decimal digit (written in decimal, of course). I've plotted all of these on a single graph, with different colours for each digit. It contains the first thousand primes.

Keep in mind that these categories of primes are not mutually exclusive. There are some trends here that are not surprising: the after the prime 5, prime numbers can only end in 1, 3, 7, or 9, since the other endings have a common factor with 10. It would probably be difficult to prove that any one function dominates another.