There are actually three "South Poles." Two are near the Amundsen-Scott South Pole Station and one is not even on the Antarctic continent! Which one is the "real" South Pole? The one that is out in the Antarctic Ocean is the south magnetic pole of the Earth. It was last located precisely at a position of 65.3 degrees south latitude and 140 degrees east longitude in 1986. Technically, it is called the south magnetic dip pole, and is at the point on the Earth where a compass needle, which is able to move vertically as well as horizontally, points straight up. The magnetic pole is due to magnetic fields that are generated deep in the Earth's core. These fields change slowly and flip from south to north on a very long time scale. For that reason, the south magnetic pole is rarely found precisely at the real South Pole of the Earth.
If you were to visit the Amundsen-Scott South Pole Station one of the first things you might do is to have your picture taken by the ceremonial South Pole. It is the one with the barber pole stripe and reflecting globe on the top. Surrounding it is a circle of flags of the nations that have signed the Antarctic treaty. This pole is where it is, because it is in a convenient location -- close to the station and within the aircraft skiway turn-around circle. But it is also not the true pole of the Earth.
The real South Pole is a couple of hundred meters beyond the ceremonial pole in the direction opposite of the station dome. It is a stake with a small brass plaque on top. Next to it is a sign that labels it as the Geographic South Pole. This one is the real thing. But if you look beyond it you see an entire string of old pole stakes. Each of them used to be the Geographic South Pole in years past. What is going on?
The reason the Geographic South Pole needs to be re-staked each year is not because the Earth's axis itself is moving*, but rather because the ice beneath the station is sliding 9.9 meters every year. When you are at the South Pole you are standing atop a nearly 3 km thick ice cap. Over long time scales, ice slowly moves down hill. In this case downhill is in the direction of the Weddell Sea, 1,400 km away. At the rate quoted above, that would mean the current South Pole marker will drop into the sea in about 140,000 years.
Don't confuse this precession with an entirely different type of wobbling -- the precession of the equinoxes, which occurs over a period of 25,600 years.