[the8472]

Since black holes are rotationally symmetric they cannot shed angular momentum via gravitational waves since the production of those requires some asymmetry[0]. But as a sibling post pointed out they can impart some of their angular momentum on objects within the ergosphere which then may escape and carry away the energy.

[0] https://en.wikipedia.org/wiki/Gravitational_wave#Sources

[andrewflnr]

I was thinking of the two-black-hole system right before merging.

[ahnitz]

Before merging, yes, they do shed angular momentum before merging. In fact, if you calculated the angular momentum of two maximally spinning black holes, you'd realize that if you could combine them, it would larger than the maximal possible spin of a single black hole with the combined mass. They must and will radiate this away as gravitational waves before merging.