I use Firefox as my default navigator on my session, using the MATE desktop (installed from the slackbuild project) with NVIDA as the X driver.

I used Firefox in 14.0, with the same configuration, without any significant issues, other than the rare annoying manner of suddenly crashing and restarting as if nothing happened. Now, using Firefox starts to be quite irritating with odd displays issues that appears suddenly, while using it (see the screenshots). Also, the font in some websites “disappear”.

I thought it was simply a bug, and I upgraded to version 24.8.1. Unfortunately, the issues are still there.
I do not have the above mentionned with Chromium (which is still too heavy for my Macbook).
What should I do?

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I use Firefox on Slackware 14.0 and 14.1, both 32 and 64 bit, but I didn't see any of the issues you describe, nor I had a Firefox crash on Slackware 14.0 ever. That might be related to your particular setup (Mate, Nvidia or other customizations) and not to Slackware itself, nor to Firefox. Hopefully someone with a setup similar to yours will be able to reproduce this. Meanwhile, you might want to remove Nvidia and your tweaks and customizations, then add them again one at a time and see when the issues show up again.