For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2\_{p,q}=H\_p(B,h\_q(F))\Rightarrow h\_{p+q}(E)\end{equation}For theories $h$ (such as ordinary homology or oriented bordism), are there results about when the sequence stabilizes (such as upper bounds on $r(p,q)$ such that $E^r\_{p,q}=E^\infty\_{p,q}$)?


What if we restrict to a class of nice spaces (such as classifying spaces $BG$ of groups)? For example, are there stability bounds for the Lyndon-Hochschild-Serre spectral sequence for group cohomology? In my limited experience, this sequence is often stable on the second page for finite abelian groups.



stability results for the Atiyah-Hirzebruch spectral sequence
The path-loop fibration for the sphere $S^n$ is: 


$$\Omega S^n \to PS^n \to S^n$$


and $PS^n$ is contractible. In singular homology, the Atiyah-Hirzebruch-Serre spectral sequence (in this case, usually just known as the Serre spectral sequence) collapses at $E^n$, and not before. That is: there is a nontrivial differential $d\_n$.


Specifically: $H\_\* \Omega S^n$ is $\mathbb{Z}$ in dimensions a multiple of $n-1$, so the SSS is concentrated in $E^2\_{jn, k(n-1)}$, where $j = 0, 1$, and $k \in 0, 1, 2, \dots$. In order for the SS to collapse to the homology of a point, it must be the case that 


$$d\_n: E^2\_{n, (k-1)(n-1)} \to E^2\_{0, k(n-1)}$$


is an isomorphism. So the SSS degenerates in arbitrarily high degrees.


Playing the same game for this fibration with any other homology theory $h$ will yield equally bad results.


