I know that finite simple groups have been exhaustively classified. Have fields (in the sense of corpora, not of vector fields) been exhaustively classified?
To expand on Makoto Kato's comment. A finite field extension of the rationals is called a number field. There are various objects associated to number fields, such as their ring of integers and ideal class groups. The ideal class group in some sense measures how badly unique factorization fails in a number field's ring of integers. One open question in algebraic number theory is whether or not there exist infinitely many number fields with trivial class group. For Galois number fields we can also talk about their Galois group. Another open question is whether or not for every finite group $G$ we can find a number field such that its Galois group is $G$. So in this sense even number fields have not been exhaustively classified.