Can't upgrade WordPress installation

I am unable to update my WordPress version to 3.4 or anything newer. When I do, I get a white page and have to back down to the older version.

Are you planning on making this a responsive version? We have an annual event, and the work always heats up right about now.. Would be nice to have the ability to upgrade. I’ll probably install a new theme for 2014 event. ridingintohistory.org