OK, Opera just released another stable version (Version:57.0.3098.106) and h.264 videos will NOT work out of the box. The problem is libffmpeg.so. You need to replace it with another that works.
Many people suggest using the Google Chrome version but this Chrome file not always works because unmatched versions. Better idea is use the version from here:
https://github.com/iteufel/nwjs-ffmpeg-prebuilt/releases/tag/0.31.5 (there are newer versions but won't work) and dawnload 0.31.5-linux-x64.zip (or the one for your system). Unzip the libffmpeg.so and replace the the one that comes with Opera (rename it as backup) that normally is in /usr/lib/x86_64-linux-gnu/opera
Close and open Opera and test it in html5test.com, check if now there is H.264 support.
Keep a copy of the libffmpeg.so downloaded from github, you may need it again when Opera is updated again without H.264 video support.

OK, Opera just released another stable version (Version:57.0.3098.106) and h.264 videos will NOT work out of the box. The problem is libffmpeg.so. You need to replace it with another that works.
Many people suggest using the Google Chrome version but this Chrome file not always works because unmatched versions. Better idea is use the version from here:
https://github.com/iteufel/nwjs-ffmpeg-prebuilt/releases/tag/0.31.5 (there are newer versions but won't work) and dawnload 0.31.5-linux-x64.zip (or the one for your system). Unzip the libffmpeg.so and replace the the one that comes with Opera (rename it as backup) that normally is in /usr/lib/x86_64-linux-gnu/opera
Close and open Opera and test it in html5test.com, check if now there is H.264 support.
Keep a copy of the libffmpeg.so downloaded from github, you may need it again when Opera is updated again without H.264 video support.