I have a fresh install of WP 3.6 hosted at 4goodhosting.com - the URL is http://middletonia.ca/blog/
The active theme is Twenty Thirteen. I use wp-admin and navigate in the left nav to Appearance > Header; in the "Select Image" area I either "choose an image from your computer", select an image and click "Upload", or I "choose an image from your media library".
Either way, the next screen shows the left nav and WordPress header but otherwise is blank. The theme header image remains unchanged. If I uploaded an image, it shows up in wp-content/uploads/2013/08 as expected. If I select an image from my media library I can browse to any of the previously-uploaded images, but again: next page is empty except for left nav and header, and the change is not applied.
Having searched the forums, I have deactivated all plugins, re-installed WP 3.6, turned on WP-DEBUG and WP_DEBUG_LOG and changed WP_MEMORY_LIMIT to 96M in wp-config.php, switched themes back and forth between Twenty Twelve and Twenty Thirteen -- no change in behaviour. I have tried all these things in many different permutations.
I do not get any PHP errors nor server errors (except 404 when I manually rename files via FTP). As far as I can tell, all permissions are correct and I can upload, edit and delete media images freely in wp-admin.
I have tried a variety of JPEG and PNG images, in sizes of 1600 x 230 (the size used by Twenty Thirteen) and other sizes. In Twenty Thirteen the cropping screen never shows up; in Twenty Twelve it did, but again the change was not applied and the default Twenty Twelve header (with no image) continued to be shown.
I have tried Chrome and Firefox on 2 different Macs (each running MacOS 10.8.4) and tried from Windows as well -- same result.
My background: lots of server and coding experience outside of PHP, some PHP experience; this is my very first WordPress site.