brandish (esp. BrE), carry, heft (esp. AmE), hold, wield
A man used to wielding an axe fought best on foot.

use
swing
sharpen

fall
The executioner's axe fell.

with an/the axe
to chop a tree down with an axe

a blow from an axe, a blow of an axe
With a few swift blows of the axe, she severed the cable.

be given, get
His prime-time TV show is likely to get the axe.

swing, wield (both esp. BrE)
Wielding the axe on the prison plan would be one way of saving money.

face (BrE)
Up to 300 workers are facing the axe at a struggling Merseyside firm.

save sb/sth from
Patients are delighted their local hospital has been saved from the axe.

fall
We were expecting bad news but had no idea where the axe would fall (= where the loss would be).