A stuck spike in a golf shoe is usually a damaged spike and will need to be replaced. Golf spike wrenches are made exactly for this purpose, but sometimes a more recalcitrant spike may require minor invasive surgery.
Gather your materials and sit on a completely flat surface with the towel underneath you and begin to spray a moderate amount of WD40 into the stuck spike or spikes receptacles. Be sure to spray enough to lube the spikes but not so much that it drips off of the spikes.
Once you have sprayed the WD40 into the spikes receptacles, move the spike wrench both in clockwise and counterclockwise manners so that eventually the spike will loosen. Work the spike wrench with a fair amount of strength because if that spike is truly wedged in bad then it will take a lot of elbow grease to release it.
After using the wrench, the spike should be at the point of ejection; however if that is not the case then it is time take another approach at removing the stuck spike. Continue on to Step 4.
If the spike is too difficult to remove using the steps above, then it is time to get the drill out. Use it to slowly drill a small hole into the spikes, but do not drill through the entire shoe. The stuck spike is a lost cause. The goal is not to damage the shoe. The drill bit you use should be large enough to allow you to insert small needle nose pliers or vise grips into the hole.
After you have finished drilling the hole, use the pliers or vise grips and place them inside of the holes that are now in the spikes. You should be able to pop them right off.