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The following result seems to be frequently quoted:

Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). Then the fiberations on $B$ with fiber $K(\pi,n)$ are classified by the space $K(\pi, n+1)$. In other words, the pull-back map $$ [B,K(\pi,n+1)]\to \{\textrm{fibrations on } B \textrm{ with }K(\pi,n)\textrm{ as fibers, up to fiber homotopy}\}$$ is a bijection.

Does anyone know a proof of this result, or a reference? Thanks a lot.

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    $\begingroup$ What's true with no hypotheses on the base is that $[B, K(\pi, n+1)]$ classifies principal $K(\pi, n)$-fibrations. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 6, 2014 at 5:58
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    $\begingroup$ Qiaochu, bundles in fact, not just fibrations, since K(A,n+1) is the classifying space of the topological abelian group K(A,n). (See e.g. my Classifying Spaces and Fibrations.) $\endgroup$
    – Peter May
    Commented Oct 6, 2014 at 12:45

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There is a very careful analysis of this question in Lemma 3.4.2, page 57, of More Concise Algebraic Topology, by Kate Ponto and myself. Assuming that $E$ and $B$ are connected, a fibration $E\longrightarrow B$ with fiber $K(A,n)$ for an abelian group $A$ is a pullback of of the path space fibration over $K(A,n+1)$ if and only if $\pi_1(B)$ acts trivially on $A$.

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