It occurred to me recently that I rarely hear about finite rings. I did a Google Ngram search to make sure this isn’t just my experience.
Why are finite groups and finite fields common while finite rings are not?
Finite groups have relatively weak algebraic structure, and demonstrate a lot of variety. Finite fields have very strong algebraic structure. Their complete classification has been known for a long time and is easy to state.
I imagine that most of the references to finite groups above have to do with classifying finite groups, and that most of the references to finite fields have to do with applications of finite fields, which are many.
You can see that references to finite groups hit their peak around the time of the Feit-Thompson theorem in 1962, and drop sharply after the classification of finite simple groups was essentially done in 1994. There’s a timeline of the progress toward the classification theorem on Wikipedia.
Rings have more structure than groups, but less structure than fields. Finite rings in particular are in a kind of delicate position: they easily become fields. Wedderburn’s little theorem says every finite domain is a field.
The classification of finite rings is much simpler than that of finite groups. And in applications you often want a finite field. Even if a finite ring (not necessarily a field) would do, you’d often use a finite field anyway.
In summary, my speculation as to why you don’t hear much about finite rings is that they’re not as interesting to classify as finite groups, and not as useful in application as finite fields.