Is there a web browser I can use safely on Mac OS 10.5.8?
You should try opera Opera 12 download it fast and works well.
Caveat: Opera only works on Macs with Intel-based processors. If you have a PowerPC chip (G5), good luck. Opera 12 won't work.
For the impatient:
You have two possibilities:
PowerPC-based Mac: TenFourFox and no Flash.
Intel-based Mac: Opera (thanks to Cybermatatu for this one!) with older Flash 10.3 (beware, Flash 10.3 is an archived version and is no longer supported nor updated with security fixes by Adobe).
Long version:
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If you use an Intel processor you can use Opera:
Opera 12
OS X Leopard (10.5), or greater: 32-bit and 64-bit Intel systems supported
Opera supports Flash Player plug-in.
If you use a PowerPC processor you may want to try TenFourFox. TenFourFox seems to be actively developed (version 17.0.3 was released Feb 16 2013). TenFourFox supports add-ons, but not plugins, that is, you can't use Flash at all.
Please notice that Flash Player 11 only supports 10.6 and newer. You can still download version 10.3 for 10.5 Intel (beware, Flash 10.3 is an archived version and is no longer supported nor updated with security fixes by Adobe).
As for other browsers:
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Firefox no longer supports PowerPC processors (PowerBook, PowerMac):
Firefox 3.6.28 is the last version of Firefox that works with Mac OS X 10.4 (Tiger) or with Mac OS X 10.5 (Leopard) on a computer with a PowerPC processor.
Firefox 16 is the latest version that supports Leopard on Intel-based Macs as Firefox 17 requires 10.6 or newer.
Apple no longer releases Safari updates for Mac OS X 10.5 (as stated here).
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Chrome doesn't support Leopard, either:
Google Chrome on Mac OS X 10.5 (Leopard) will stop receiving any updates following Chrome 21.
I have had some success in a Flash-less world by using Safari, and enabling the Develop menu. Then when I get to a page that requires Flash, I can set the User Agent to report as an iPad and I will get videos streamed in a QuickTime player rather than Flash-based.