  We investigate the possible self-intersection numbers for sections of surface
bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the
base genus h are positive, we prove that the adjunction bound 2h-2 is the only
universal bound on the self-intersection number of a section of any such genus
g bundle and fibration. As a side result, in the mapping class group of a
surface with boundary, we calculate the precise value of the commutator lengths
of all powers of a Dehn twist about a boundary component, concluding that the
stable commutator length of such a Dehn twist is 1/2. We furthermore prove that
there is no upper bound on the number of critical points of genus-g Lefschetz
fibrations over surfaces with positive genera admitting sections of maximal
self-intersection, for g at least two.
