What (if any) regularly scheduled airline flights pass over Antarctica?

As Mark noted, no scheduled flights pass directly over Antarctica. As to which goes closest, I think the Qantas flight QF17 / QF18, between SYD (Sydney) and EZE (Buenos Aires), is a good candidate.

Mark said BA to Auckland would be the southernmost flight, but the post he quoted is from 2005 and Qantas started the QF17/18 service in November 2008. Also, looking at the (great circle) map, it only makes sense that BA - Sydney would go farther south than BA - Auckland.

^{Image from Great Circle Mapper}

In this post at Sydney Airport Message Board I found route details for QF17, from its inaugural service, apparently written by one of the pilots. The southernmost latitude reached on the SYD-EZE flight was 63°. And looking closer, on the return leg (EZE-SYD), they flew much closer to Antarctica, at 72°, at one point passing "over Thurston Island and Cape Flying Fish, part of continental Antarctica".

Quoting some interesting commentary:

(Note the deep south latitudes on this sector. Being handed over to McMurdo Centre ('Mac Centre') on HF was certainly a highlight).

[...]

Ops are currently limited to a maximum of 72 deg South, which provides a small buffer against overflying Antarctica proper (save for the low-lying cape/island we overflew). Going further south entails similar considerations to our Antarctic charter flights (eg. Polar survival equipment, which requires the removal of a number of seat rows). An analysis is currently being undertaken by QF on the concept of going to 80 deg South, which would allow westbound flights to take advantage of significant tailwinds there. Going down to 80S (versus staying further north) can make the difference between an overall wind component of 40kts headwind and 10kts tailwind.

So, at least at the time of that writing (Nov 2008), 72 degrees S was the southernmost limit for regular Qantas flights. The post also has some interesting photos, e.g. of the (1st leg) route plotted on a "Jeppesen South Pole Plotting Chart".

Incidentally, I took the QF18 from BA to Sydney last spring. I found it pretty interesting and was actually wondering at the time whether it would pass over Antarctica or just near it. Here are some photos I took:

Looking left (towards the South Pole):

Looking right (north):

According to Airliners.net, Aerolineas Argentinas operates the world's southernmost (scheduled) commercial route in the world, from BUE (Buenos Aires) to AKL (Auckland), which flies in about 50-55 deg South of the equator.

(As it happens I've taken this flight, which is a nice bit of knowledge for me :))

Second is probably JHB to SYD (Johannesburg to Sydney) with Qantas or South African Airlines, which goes down to 45S.

it turns out that commercial flights are actually NOT allowed over Antarctica for security reasons (see Mt Erebus disaster), aside from Boeing who in 2003 went from SYD to GIG (Sydney to Rio de Janiero), crossing over the south pole in the process.

This has been mostly brought in due to the fact that most of Antarctica is out of ETOPS range, and this is also why LAN Chile and Aerolineas Argentinas fly the Southern Pacific with 4-engine planes.

There are sightseeing flights, but not regularly scheduled commercial routes over the continent any longer.

I flew from SYD to SCL at the start of last year (Jan 2013) specifically to go on a 20-day Antarctica cruise. We got as far south as just shy of 65 degrees on the cruise.

On the flight back (Qantas QF28) from SCL to SYD we got so far south that I was able to take some photos of tabular icebergs from the aircraft. Around the same time I managed to grab one of the pilots who was walking down the aisle to have a chat (I'm an aircraft engineer and we probably ended up chatting about half an hour). I specifically grabbed him aside in the first place to ask him what lattitude we were flying at, and he said we were at 70 degrees! Further south than the 20 day cruise took us, plus inside the Antarctic circle!

To expand on Mark's JHB-SYD suggestion, the Johannesburg to Sydney route generally does not go far enough south to sight Antarctica because it can take advantage of the winds in the Roaring 40s. However, SYD-JHB will sometimes travel within sight of Antarctica (if not actually over it for obvious reasons) to avoid those same winds. The below photo shows us travelling over the sea ice looking south towards Antarctica.