is it for using on live site or not?

There is README file: http://cvs.drupal.org/viewvc.py/drupal/contributions/modules/dtools/wsod...
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WSOD module should be used only, when you have some problems with your stability and you have recently White Screen of Death to diagnose it.
You should not have problems with using it on live as well, but the only problem is that module using register_shutdown_function to make some serveral diagnostic tests for problems, and some contributed modules are not written well for this case.
#337615: require_once failure on cron with modules that have a register_shutdown_function
#375153: Failed opening required pathauto.inc on line 74
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If you mean bench_chart module, as every statistic module, it's slowing down your website by calculating execution times. It's good to check for some problems with your website (why is so slow and which module is slowing down your website the most), and after making some optimization, is good to disable it.

It helped me a lot in some cases.
In example:
#349375: Sometimes update_help() execute ages, Drupal freezing
#222073: Update Status causing Drupal to run very slowly
When my website was loading few minutes!
See the chart: http://tinyurl.com/9rbv5d

Sorry to reopen a closed issue, but I wanted to add my experience. This module absolutely killed my site's load when I enabled it on the live site. So I would say, definitely *not* for use on a live site, at least one that is frequently visited (& which may have PHP errors).

Added new option in settings page of wsod module to not checking for wsod's on every page (which was slowing down the website). As default this option is not active, but it could be activated on test servers where you want to monitor each WSOD in watchdog.

Fixed: #523278: Extreme bug: bench_chart kills off Drupal search indexing!
If any of you using search indexing, consider to download laster dev.