A few ideas: Do you have the latest Adobe flash update, and did you get it by going to the Adobe website, and downloading it from within the Opera browser. If not, that's what you need to do. I believe the latest version is 18.104.22.168. Also, if you're using an ad-blocker, like AdBlock plus (but I guess it could be another), perhaps you have blacklisted YouTube accidentally. So you might want to take a closer look at your adblocker. Also, check you're firewall, and see if that's doing something adverse. You could try deactivating all of your extensions and see if that makes a difference.
Here are some links that might provide help, even though the poster may have been using a different browser. There seems to be a commonality to YouTube problems.