For a fibration $F \to E \to B$ with $B$ simply connected and field coefficients $k$: $E_2^{p,q} = H^p(B; H^q(F; k)) \Rightarrow H^{p+q}(E; k)$

This computes cohomology of the total space from the base and fiber.

For a continuous map $f: X \to Y$ and sheaf $\mathcal{F}$ on $X$: $E_2^{p,q} = H^p(Y, R^qf_*\mathcal{F}) \Rightarrow H^{p+q}(X, \mathcal{F})$

This relates sheaf cohomology on $X$ to derived push-forward on $Y$.

For composable functors $F: \mathcal{A} \to \mathcal{B}$ and $G: \mathcal{B} \to \mathcal{C}$, if $F$ sends injectives to $G$-acyclic objects: $E_2^{p,q} = R^pG(R^qF(M)) \Rightarrow R^{p+q}(G \circ F)(M)$

This computes derived functors of compositions.

For a group extension $1 \to N \to G \to Q \to 1$ and $G$-module $M$: $E_2^{p,q} = H^p(Q, H^q(N, M)) \Rightarrow H^{p+q}(G, M)$

This is the Grothendieck spectral sequence specialized to group cohomology.

For a pullback square of spaces, the Eilenberg-Moore spectral sequence computes cohomology of the pullback from Tor groups: $E_2^{p,q} = \text{Tor}^{H^*(B)}_{p,q}(H^*(X), H^*(Y)) \Rightarrow H^{p+q}(X \times_B Y)$