spectral sequences in number theory

I posit the following example, in response to your ambiguous question:

The coniveau spectral sequence seems to play an important role in 'arithmetic geometry'. One instance is in class field theory for schemes:

From W. Raskind's nice survery article "Abelian class field theory of arithmetic schemes" [AMS, 1992, pgs. 100-101]:

Let $X$ be an arithmetic scheme, $n>0$ invertible on $X$. Then there is a coniveau spectral sequence (in the etale site):

$$E^{p,q}_1 = bigoplus _{ xin X^{p} } H^{q-p} (k(x), ;mathbb{Z}/n ; (j-p)) Rightarrow H^{p+q} (X, mathbb{Z}/n ;(j)) $$

Without going into more details, this sequence plays an important role in defining a reciprocity map from a class group of $X$ to abelian fundamental groups.

That's all I will say for now in hopes that the above provides for motivation to delve further into studying coniveau, etc.

Finally, one of the best articles I have seen on coniveau is by Colliot-Thélène, Hoobler, and Kahn, "The Bloch-Ogus-Gabber theorem" which can be found at:
http://www.math.jussieu.fr/~kahn/preprints/prep.html

It might be nice to have others' remarks/comments on coniveau, but I don't have any precise questions yet.