I've just spent an hour figuring out a solution. The problem is that Opera's 'Local State' file has become either corrupted or unsuited to the new version of Opera.
Close Opera. Close all running instances of the browser.
Browse to your local Opera profile folder. In Linux this is under /home/[username]/.config/opera/. I have no idea where this Opera folder is in Windows or Mac. You will need to look for it. I think someone mentioned it earlier in this discussion thread.
Make a backup of the above 'opera' directory. This directory stores all your personalisations of the browser and stores any saved sessions; then
Rename the file 'Local State' to 'xLocal State'
Your Speed Dials should display again; if not, they will be stored in Bookmarks > Other Speed Dials and from here you can select and drag them back into Speed Dials.
If the above solution causes your browser to lose its settings, just close Opera and restore the backup you made of the 'opera' directory in step 2.
The problem is that the Local State file prevents the Speed Dials folder from loading. It is as though Opera looks in the wrong place for the Speed Dials data.
Hope this helps,