TITLE: Which axioms of ZFC or PA are known to not be derivable from the others?
QUESTION [16 upvotes]: Which, if any, axioms of ZFC are known to not be derivable from the other axioms?
Which, if any, axioms of PA are known to not be derivable from the other axioms?

REPLY [7 votes]: This is a community wiki answer to gather references. Please feel free to edit it. 

Abraham Robinson. On the independence of the axioms of definiteness (Axiome der Bestimmtheit), Journal of Symbolic Logic, Volume 4, Number 2 (1939), pp. 69-72.
Elliott Mendelson, Some Proofs of Independence in Axiomatic Set Theory, Journal of Symbolic Logic, Volume 21, Number 3 (Sep., 1956), pp. 291-303. MR0084463 (18,864c).
Paul Cohen. The independence of the continuum hypothesis, Proc. Nat. Acad. Sci. U.S.A., Volume 50, Number 6 (1963), pp. 1143–1148. MR0157890 (28 #1118); and The independence of the continuum hypothesis. II, Proc. Nat. Acad. Sci. U.S.A., Volume 51, Number 1 (1964), pp. 105–110. MR0159745 (28 #2962).
Alexander Abian and Samuel LaMacchia. On the consistency and independence of some set-theoretical axioms. Notre Dame Journal of Formal Logic, Volume 19, Number 1 (1978), pp. 155-158. MR0477290 (81e:04001).
Greg Oman. On the axiom of union, Arch. Math. Logic, 49 (3), (2010), 283–289. MR2609983 (2011g:03122). See also this MO question, and here. (I do not know of an original reference for the fact that $\mathsf{ZFC}-\mathrm{Union}$ does not suffice to prove the existence of infinite unions. The paper includes the usual proof of this fact, and clarifies precisely which unions can be proved to exist in this theory: $\bigcup x$ exists iff $\{|y|\colon y\in x\}$ is bounded.)