I've done more investigating and have interesting results. I installed Opera 9.64 on my Athlong 64 computer; Flash had already been updated to the latest version. This time I used the Slackware 11-and-later version offered at opera.com instead of my usual static tar.gz version. Flash continued to work with the latest Opera on that computer. I use the proprietary ATI driver on it.
Starting Opera from a terminal prompt on my other computers, I get error messages if I have plugins activated. The error messages aren't the same. I'm at the 1.3 GHz Celeron as I write this, and this is the error message:
Not GTK2 toolkit (got 0).
Opera itself works fine, but Flash does not. When I originally installed VL6 Standard Deluxe Opera 9.63 and Flash were working, but somewhere along the upgrade line Flash stopped working. This computer uses the xorg radeon driver. Flash works fine on SeaMonkey and Firefox.
My Turion 64 also has Flash non-working with Opera. The error message is different. I'll post it tonight when I'm using that computer. That computer uses the proprietary NVidia driver.
Can anyone explain what "Not GTK2 toolkit (got 0)" means and what I can do about it? All three computers have just about the same software, so how GTK2 toolkit disappeared is a mystery to me.
I should also mention that I removed Opera entirely from the Celeron computer and reinstalled with the same Opera-provided Slackware version as I used on the computer where Flash and Opera are working together. Same results--Flash doesn't work on the Celeron. I also reinstalled the latest Flash, which didn't help, either. I don't know what else to try.