Not only can you camp on BLM administered lands, but you can include US national forests as well. We've never had any safety issues camping on public lands other than an odd run-in with a fellow out in the desert outside Quartzsite, AZ.
BLM has developed campgrounds in some pretty great places in the west. The following link shows a map of BLM's numerous recreation areas. You might even notice one in Florida. Zoom out on the map to see Alaska too. Clicking on the green and yellow dots reveals the name, but I find you have to search for the name to find details...
BLM Recreation Areas