Even when they have been dead for years, tree stumps are solid and stubborn and can be hard to remove. However, you don't need any heavy equipment to safely and completely remove most tree stumps. The method you use depends on the area the stump is in and what is safe to use, but there is an option for almost any situation.
Cut the Stump Out
Even if you're looking at a medium-sized stump, it can be removed by hand if you have a sturdy shovel and a saw. This is the fastest method of stump removal as you can remove the whole stump all at once after you've finished digging.
Start digging around the trunk's circumference about a foot from the stump. Dig straight down until you start hitting the roots. Dig around and under the roots as best you can. You don't need to expose the ends, just the roots close to the stump. If you're having trouble digging around the roots, a spade comes in handy here. Use the saw to cut the roots away from the base of the stump. You want to leave the stump with nothing holding it into the ground.
Once you've dug far enough to start reaching under the stump, you can try to pry it or pull it out of the ground. Alternatively, if you have a vehicle and some sturdy rope nearby, you can attempt to pull it out that way. If you feel too much resistance, dig around the stump some more rather than applying too much pressure.
Use an Herbicide
An herbicide will kill a stump over the period of a few weeks. You can still use this on older stumps, though you'll need to take an extra step.
For fresh stumps (trees cut down within the past half a year or so), drill a number of holes into the stump about four to five inches deep. Pour an herbicide of your choice into each of the holes. Expect to let this sit for a few weeks; after this time the stump should have started decaying and softening, making it easier to remove.
If the stump is older, you will first need to make a new cut about an inch down to expose newer wood. Once you've cut an inch off the existing stump, you can then drill the holes and use the herbicide.
Once the herbicide has done its job, it will be much easier to cut out using an axe or shovel; dead wood is much easier to break apart.
Burn the Stump
If you don't want to use chemicals, or want to try something that works a little faster, you can try burning the stump out of the ground. Before you try this, you should make sure of two conditions: 1) That your stump is old and dry and that there is no other plant life around the stump that can catch fire, and 2) that there are no fire codes in your area prohibiting you from burning out your stump.
Clear the area around the stump and dig away some of the dirt surrounding the stump to expose as much of the wood as you can. Place dry wood around the stump so that the stump is in the center of the wood pieces. Finally, light the wood with the aid of fuel oil or kerosene and let it burn.
You will need to supervise this process, potentially for a few hours. Something as solid as a stump won't burn easily, so you may need to add wood repeatedly to keep the fire going. Eventually, the stump should catch fire and slowly burn until there is mostly ash left. Let the ashes sit for at least a day before you remove them; the ashes can retain heat for hours and you risk getting burned. For assistance, talk to a professional like DJ's Tree Service.