One example is a construction that is often used in the passage from smooth projective varieties to arbitrary varieties. There are various variants of this:
- For a proper variety $X$, take a hypercovering $X_\bullet \to X$ by smooth projective varieties (using alterations plus the inductive procedure of SGA 4$_{\text{II}}$, Exp. V$^{\text{bis}}$, §5). Then there is a hyperdescent spectral sequence $$E_1^{p,q} = H^q_{\text{ét}}(X_p,\mathbf Q_\ell) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$
- For a smooth variety $X$ with a smooth compactification $\bar X$ whose complement $Z = \bar X \setminus X$ is a simple normal crossings divisor $\bigcup_{i \in I} Z_i$, there is an excision spectral sequence $$E_1^{p,q} = \bigoplus_{\lvert J \rvert = p} H^q_{\text{ét}}\Big(\bigcap_{i \in J} Z_i,\mathbf Q_\ell\Big) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$
- These methods can be combined: as I explain in [vDdB20, Thm. 6.6], for an arbitrary separated finite type $k$-scheme $X$, one may construct a simplicial variety $X_\bullet$ with smooth projective components $X_i$ and a spectral sequence $$E_1^{p,q} = H^q_{\text{ét}}(X_p,\mathbf Q_\ell) \Rightarrow H^{p+q}_{\text{ét}}(X,\mathbf Q_\ell).$$ It has been suggested to me that one should be able to prove this using Voevodsky motives, but I don't know a reference. (My paper is written in the more classical language of Chow motives.)
There are also versions for singular cohomology if $k = \mathbf C$. For instance, the first example above is used to define the mixed Hodge structure on the cohomology of an arbitrary $\mathbf C$-variety.
Each of the above examples degenerates on the $E_2$ page for weight reasons: any $E_r^{p,q}$ is pure of weight $q$, so the differentials are forced to be $0$ once the source and target live in different rows. The observation that you only need the $E_1$ differentials is one of the key points of [vDdB20].
(To make this weight argument precise, either reduce to a finite type situation over $\operatorname{Spec} \mathbf Z$ and use Frobenius eigenvalues, or (if $\operatorname{char} k = 0$) reduce to a finite type situation over $\mathbf Q$, choose an embedding into $\mathbf C$, and use Hodge theory.)
References.
[vDdB20] R. van Dobben de Bruyn, The equivalence of several conjectures on independence of $\ell$. Épijournal Géom. Algébrique 4, Art. nr. 16 (2020). ZBL1460.14053.