For which codecs/containers does Windows Media Player now have inbuilt support with Windows 8?

I don't think there are any.

In fact, in Windows 8, the Windows Media Player does not include any DVD codecs. You cannot play any DVDs, due to licensing issues. Microsoft has said:

Windows Media Player will continue to be available in all editions, but without DVD playback support. For optical discs playback on new Windows 8 devices, we are going to rely on the many quality solutions on the market, which provide great experiences for both DVD and Blu-ray.

Source: Microsoft ejects DVD playback from Windows 8

Personally, I prefer VideoLAN (free), and it appears to play anything I throw at it. Because it is open source, it also means there are no commercial subtle hints to "buy stuff from our store", which is good.

You can check with sherlock - which shows and lets you save an output of what codecs you have installed. The following are the codecs that Windows 8 comes with. There may be additional ones with Windows Media Center, but I wouldn't consider these built in. While the built in DVD codec was axed, there are other codecs built in.

 DirectShow Runtime.
 Cinepak® Codec
  XDSCodec & Encypter/Decrypter Tagger Filters.
 DirectShow Runtime.
 Intel Indeo(R) Video YUV Codec
 Microsoft RLE Compressor
 Microsoft Video 1 Compressor
 DirectShow Runtime.
 Windows Media MPEG-4 Video Decoder
 Windows Media MPEG-4 S Video Decoder
 Windows Media Screen Encoder
  PTFilter & Encypter/Decrypter Tagger Filters.
 VBI Surface Allocator Filter
 DirectShow Runtime.
 Windows Media Screen Decoder
 Windows Media Video Decoder
 Windows Media Video Encoder
 Windows Media Video 9 Encoder