This is very similar to problems described at #560558: Where is the documentation? supercron.php throws error in browser
Have installed supercron because a drupal-6.15 site has no search indexing (it worked on a development site on a web host with PHP5.2.x but has failed since moved to an on-site production machine with PHP5.3.0).
Am running PHP5.3.0, so patched according to #1010226: Saving settings form not PHP 5.3 Compatible.
Then could load /admin/settings/supercron to get list of parts of cron to run.
The Last called column shows nothing for search (or for anything after).
The Output and Exceptions columns shows (empty) everywhere.
If I run search only using /admin/settings/supercron/call/search/1 it seems to stall for a bit then a server message comes back "because the server unexpectedly dropped the connection. This sometimes occurs when the server is busy. Wait for a few minutes, and then try again."
/admin/settings/supercron/invocation gives White Screen of Death.