Debian is the universal operating system, with many packages pre-compiled for many architectures.
InstallingDebianOn article for BeagleBone: Debian wiki instructions
All BeagleBone Black Rev C boards are now shipping with Debian images (started May 5, 2014). See the Latest Images Page for links to those images. For instructions on rebuilding those images, see http://beagleboard.org/source. Issue reports should be filed at http://bugs.elinux.org/projects/debian-image-releases. Additional test images and on-going development can be found on the BeagleBone Black Debian wiki page.
Latest builds for BeagleBone, BeagleBone Black and all BeagleBoard Compatible BeagleBones can be found at http://builds.beagleboard.org.
For Debian images for boards other than BeagleBone, check out http://elinux.org/BeagleBoardDebian.