If you're familiar with the world of embroidered patches you've probably noticed quite a few utilizing logos from the US military. This is pretty common practice, especially when you consider how popular military patches are. As a matter of fact, it's not uncommon for patch dealers to submit their own military-based design to a manufacture who will then create a certain number of wholesale patches for the reseller.
However, designers and manufacturers need to be aware of some things regarding the use of government images. By the letter of the law all government images are in the public domain due to the fact that the government is an institution of the people. Yet some of those images do have restrictions that need to be taken into consideration.
For example, while artists and patch manufacturers are free to use military logos and slogans they must do so within certain parameters. They cannot create anything that is blatantly offensive, potentially dangerous to military service men and women, and so on. All uses of military logos must be tasteful and honoring to the institution from where they are taken. When it comes to certain agencies like NASA, greater restrictions are in place.
NASA logos and seals are under the control the agency's director. According to federal an artist could not duplicate a NASA logo right down to the very last detail without written permission to do so. He could create an image that is similar as long as there are enough differences to clearly distinguish it from the original.
If you're planning to use government images for your own patches make sure you check before doing so. In most cases you won't have a problem. But where restrictions exist you could get yourself into serious trouble if you don't seek permission first.