Depends on the roots' relative size:
Axe/Hatchet, hand-saw, long-pike/wrecking-bar, loppers, mattock/pick-axe, & pointed-shovel/spade
are the normal/old-hand-tools for the job.
Power (chain/circular) saws are VERY DANGEROUS to try to use on roots.
(because of the kick-back risks)
A rototiller can break up smaller roots.
A (power) plough/plow, stump-cutter, or trencher is required to break up larger roots.