Statistics on the frequency of usage of RSA key public exponents
I couldn't find any references to any statistics, however:

All Fermat primes $${3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … }$$ could be used while regarding the use a proper padding scheme. There is no known weakness for any short or long public exponent for RSA, as long as the public exponent is "correct" (i.e. relatively prime to p-1 for all primes p which divide the modulus).

So why is $ e = 65537$ used most commonly?

Using $e=65537$ (or higher) in RSA is an extra precaution against a variety of attacks that are possible when bad message padding is used. But it's not too large so that it would greatly impact performance speed ($e = 3$ is around 8x faster than $e = 65537$).

So in short: $e = 65537$ is most commonly used as a comprimise, because it's reasonably fast and secure.

Related answers for more details:


Impacts of not using RSA exponent of 65537, answer by fgrieu
Should RSA public exponent be only in {3, 5, 17, 257 or 65537} due to security considerations?, answer by Thomas Pornin