In 2017, Miller computed the character tables of S_n for all n up to 38 and looked at various statistical properties of the entries. Characters of symmetric groups take only integer values, and based on his computations, Miller conjectured that almost all entries of the character table of S_n are divisible by any fixed prime power as n tends to infinity. More recent computations of Miller and Scheinerman suggest that the density of zeros appearing in the character table of S_n tends to zero as n tends to infinity, and that the vast majority of these zeros are of a certain special type. Soundararajan and I were able to prove Miller's conjecture on the divisibility of entries of character tables, but it's still not known whether almost all entries are nonzero, though we can explain some other phenomena observed in the data generated by Miller and Scheinerman. I will describe the ideas going into our arguments and discuss some related open problems.