TITLE: Finite local rings
QUESTION [9 upvotes]: There is some classification of finite commutative local rings. For example how many not isomorphic finite local rings with the same order $p^k$ and the same residue field $\mathbb F_p$ exist?

REPLY [5 votes]: Corbas and Williams answered this question for local rings with p^k elements, where k is at most 5.  See "Rings of Order p^5 Part I. Nonlocal Rings" and "Rings of Order p^5 Part II. Local Rings". Both papers are in Journal of Algebra, vol. 231; the first on pages 677-690 and the second on pages 691-704.