What apprenticeship programs exist for service members and/or veterans? 

Apprentices and Career Seekers

United States Military Apprenticeship Program (USMAP) is a formal military training program that provides active-duty Army, Navy, Marine Corps, and Coast Guard service members the opportunity to improve their job skills and to complete their civilian apprenticeship requirements while they are on active duty. The benefit to the service is a cross-trained service member with hands-on experience in all aspects of their MOS/rating.     

The DoD SkillBridge initiative is a separate program that allows transitioning service members, within six months of separation, to participate in civilian job and employment training, including apprenticeships and internships.  Additionally, the training must offer a high probability of employment and be provided to the Service member at little or no cost.