Replacing a rear wheel spoke requires that you remove the freewheel or cassette, the set of sprockets comprising your rear gears. Most 10-speed bikes manufactured since the mid-1980s use cassettes; older 10-speeds use freewheels. Freewheels screw onto the rear wheel hub, while cassettes slide onto a freehub and lock into place with a lockring. To identify which you have, take the rear wheel off the bike and spin the sprockets backward. If the sprockets rotate independently of the center of the sprocket cluster, you have a freewheel. If the center and sprockets rotate together, you have a cassette.
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Loosen the rear wheel's axle locknuts or quick release skewer, deflate the tire and pull the wheel from the bike.
Place the cassette or freewheel remover into the center of the sprocket cluster. For cassettes, place the chain of the chain whip on the middle sprocket, with the handle on the right side and the chain loop touching the sprocket teeth. For freewheels, put the axle locknut or quick release skewer back on the axle and tighten it to hold the freewheel remover in place.
Use the adjustable wrench to turn the freewheel or cassette remover counterclockwise, while turning the chain whip clockwise, if applicable. You can more easily remove a freewheel by placing the remover in a vise, putting the freewheel on it, and turning it counterclockwise until the freewheel comes off. Put all the parts of the cassette or freewheel aside.
Remove the tire and tube from the wheel, using the tire levers to lift the bead of the tire over the rim and pulling the tire and tube away starting at the valve stem. Pull the rim strip from the inside of the wheel.
Try the different sizes of the spoke wrench on the spoke nipple until you find one that doesn't rotate independently of the nipple when turned. Look down on the spoke nipple from above and turn the spoke wrench clockwise until the spoke comes out of the nipple.
Set the spoke nipple aside. If a piece of the spoke is still attached to the hub, change its angle until you can push or pull it out, using pliers if needed to bend the spoke.
Slide the new spoke into the hub. The flattened flange should face the opposite direction of the spokes on either side of it in the hub. Bend the spoke if needed so that it doesn't slide into small spaces between spokes on the other side of the hub; you need room to rotate it into place.
Lace the spoke between adjacent spokes, mimicking the lacing pattern of the other spokes.
Insert the spoke nipple into the rim hole and use the spoke wrench to tighten the spoke until it has the same tension as other spokes coming from that side of the hub. Bicycle Tutor recommends plucking the spokes and adjusting the new spoke's tension to make the same sound.
Reattach the cassette or freewheel, using the remover to tighten it into place. Put the rim strip, tube and tire back on the wheel.