The difference between algebraic geometry and algebraic topology is that in AG, you usually hope that your spectral sequences degenerate immediately at the $E_2$ page. In AT, you often have to live with spectral sequences which never degenerate. I’m curious about spectral sequences with “intermediate” behavior.

Question 1: What are some examples of spectral sequences which have exactly one page of nonzero differentials?

Most interesting spectral sequences are sufficiently natural in their inputs that the first “globally” nonvanishing differential is some kind of cohomology operation. In a particular instance of the spectral sequence, the first nonvanishing differential might be this globally first nonvanishing differential, or it might come later.

Question 2: In these examples, does the unique nonvanishing differential coincide with the “globally first nonvanishing differential”?

For example, in the Atiyah-Hirzebruch spectral sequence for Morava $K$-theory $H^s(X;K(h)^t) \Rightarrow K(h)^{s+t}(X)$, the first globally nonvanishing differential is (a scalar multiple of) the $h$th Milnor primitive $Q_h : H^s(X) \to H^{s+2^p-1}(X)$ (at $p=2$). When $X = \mathbb R \mathbb P^\infty$, one can work out that this is the unique nonvanishing differential, so it’s an example answering to Question 1, and the answer to Question 2 in this case is “yes”.