The easiest and most efficient way is to rent a stump grinder. It'll be the best $50-$100 you will spend.
Depends on how many you have to get rid of.
I've dug out quite a few over the years and , although there are exceptions, most came out with the one method.
When I cut the trees I left perhaps 4 or 5 feet of trunk to use as a lever then dug around the stump using a mattock, spade, and crowbar. As the roots were exposed I cut through them, removing them whenever possible. When there was enough movement in the trunk I either attached a block and tackle (that's the easy way), or just muscled up to it and shoved (that's the hard way) until the tap roots were exposed. Then out with a trusty axe (or the back of the mattock) and hacked through them like the base of Jack's beanstalk.
Of course, there are exceptions and sometimes you need to be creative, but that's a whole other story.
Dig it out of the ground. Duhh lol
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