  We study Brauer's long-standing $k(B)$-conjecture on the number of characters
in $p$-blocks for finite quasi-simple groups and show that their blocks do not
occur as a minimal counterexample for $p\ge5$ nor in the case of abelian
defect. For $p=3$ we obtain that the principal 3-blocks do not provide minimal
counterexamples. We also determine the precise number of irreducible characters
in unipotent blocks of classical groups for odd primes.
