We construct a triangulated analogue of coniveau spectral sequences: the motif of a variety over a countable field is decomposed (in the sense of Postnikov towers) into the twisted (co)motives of its points; this is generalized to arbitrary Voevodskys motives. To this end we construct a Gersten weight structure for a certain triangulated category of comotives: the latter is defined to contain comotives for all projective limits of smooth varieties; the definition of a weight structure was introduced in a preceding paper. The corresponding weight spectral sequences are essentially coniveau one; they are $DM^{eff}_{gm}$-functorial (starting from $E_2$) and can be computed in terms of the homotopy $t$-structure for the category $DM^-_{eff}$ (similarly to the case of smooth varieties). This extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the cohomology of a smooth semi-local scheme is a direct summand of the cohomology of its generic fibre; cohomology of function fields contain twisted cohomology of their residue fields (for all geometric valuations). We also develop further the general theory of weight structures for triangulated categories (independently from the motivic part of the paper). Besides, we develop a certain theory of nice pairings of triangulated categories; this subject seems to be new.
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