The latest 4GB Debian wheezy image currently advertised on the website (http://beagleboard.org/latest-images) has an issue with not properly launching the LXDE X11 desktop:
http://builds.beagleboard.org/images/master/08132bf0d0cb284d1148c5d329fe3c8e1aaee44d/bone-debian-7.9-lxde-4gb-armhf-2015-11-12-4gb.img.xz
Specifically, lxpanel doesn’t start automatically, and so you don’t get the task bar, widgets, desktops, etc that come with it launching. The File Manager doesn’t launch either in the intended way, which means that no desktop wallpaper image will be set.
The reason is that the file /etc/xdg/lxsession/LXDE/autostart erroneously can be read only by root. This can be fixed easily (once you know that this is the problem, which took be countless hours and running down rabbit holes 
$ sudo chmod a+r /etc/xdg/lxsession/LXDE/autostart
I can’t say anything about other images. They may or may not have the same issue, but if they do, the same fix will in all likelihood apply.
Presumably this will be correct in future images because the underlying bug has already been fixed thanks to Robert [1].
-hilmar
[1] https://github.com/RobertCNelson/omap-image-builder/issues/68