Single Parents are not allowed to enlist in the US Military, period. Except for the Army National Guard, waiver approvals are very, very, very rare, and most recruiters won't even submit one. In the "old days," some recruits would try to get around this restriction by giving up legal custody of their child(ren) until after basic training and job school, but the military has wised up to this practice.
For example, in the Marine Corps, one must give up legal custody (by court order) of their child(ren), and then wait one year or more before being eligible for enlistment. For Navy enlistments, the waiting period is six months and the court-order must make it very plain that the transfer in custody is permanent.
In the Army and Air Force, single member parent applicants who, at the time of initial processing for enlistment, indicate they have a child or children in the custody of the other parent or another adult are advised and required to acknowledge by certification that their intent at the time of enlistment was not to enter the Air Force/Army with the express intention of regaining custody after enlistment. These applicants must execute a signed statement testifying they have been advised that, if they regain custody during their term of enlistment, they will be in violation of the stated intent of their enlistment contract. They may be subject to involuntary separation for fraudulent entry unless they can show cause, such as the death or incapacity of the other parent or custodian, or their marital status changes from single to married.
The military's refusal to accept single parents for enlistment is a valid one. The military is no place for a single parent. Due to a divorce, I spent the last six years of my military career as a single parent, and it is the singularly most difficult thing I have ever done in my life. In the military, the mission always comes first. Absolutely no exceptions are made in assignments, deployments, duty hours, time off, or any other factor for single parents. Single parents in the military are required to have a nonmilitary person (in the local area) on call at all times, 24-hours-per-day, seven-days-per-week, 365 days-per-year, who will agree (in writing) to take custody of their child(ren) at no notice, in the event that the military member is deployed or called to duty. Failure to comply with these "Family Care Plans" can (and does) result in an immediate discharge.
In general, an applicant who has joint physical custody of a child by court order or agreement, and the applicant does not have a spouse, he/she is considered a "single parent." If local or state court allows modification, if the other parent assumes full custody, the applicant is usually qualified for enlistment.
In the Army National Guard, a single parent may enlist, if they receive a waiver
from the State Adjutant General of the state that individual is enlisting.