Version PHP5.2 allows my v1.5 ZenCart to work perfectly - when I change my PHP version to PHP5.3 - ZenCart 1.5 does not work.

Any ideas?
Zen Cart 1.3.9h, 1.5.0, and 1.5.1 all work out of box in my test environments under PHP 5.2 and 5.3. I have a couple test environments running under PHP 5.4 - and after disabling strict error reporting I have not had any major issues.

Check all your plugins and modifications to see if they are PHP 5.3 safe. The link was intended more for people writing plugins, but will not be a bad starting place. The instructions in the "how to fix blank pages" should also help - keep in mind however FATAL PHP errors if present can sometimes show up in the web server's error log (if enabled) instead of the Zen Cart debug logs.