Do any flights go over the South Pole?
Please forgive me for asking if you are a "flat earth truther".  We sometimes get those numpties passing by trying to justify their theories (especially the one about Antarctica being a ringed ice wall with the military guarding it to make sure no-one learns the secret).

There have been many flights over the geographic South Pole, the first documented one being in 1935 by Lincoln Ellsworth and Herbert Hollick-Kenyon.  See also the first flight over Antarctica.

Pan Am flight 50, using a Boeing 747SP, also flew over both poles.

No scheduled commercial flights over fly the South Pole since there are no great circle routes in use which do so.

Flights from New Zealand to South Africa would fly over Antarctica but no airline currently flies that route.

As to problems with the compass, modern aircraft have no problems since the primary navigation source is GPS.  If you were using a compass only, as you got close to the magnetic South pole (remember, this is not even in Antarctica), the compass would become increasingly unreliable.  Simply maintain a straight course though, over the pole, and the compass will become increasingly reliable.  However, navigating over the magnetic poles using only a compass is the preserve of explorers and the fool-hardy.

This document contains more technical detail regarding navigation.  It talks mainly about the North magnetic pole but is applicable to both.

A couple of other considerations when asking about commercial flights is the need for ETOPS certification and the need to carry special survival equipment when flying south of latitude 72 degrees which is very considerable and would require the removal of revenue seats to carry it.  

There is a big difference between the nearest suitable diversion airport flying over the Arctic and flying over the Antarctic.