How do an unmarried woman under 21 also is a parent of a 5months American born baby with an expired visa apply for permanent residence or asylum

Based upon the information provided you do not appear to be eligible for a green card. Having a United States citizen child creates no path to residence unless the child is at least 21 years of age. You could file for asylum. You file for asylum by completing a Form I-589 and filing it with USCIS. To be eligible for asylum, you must have a well-founded fear of persecution on account of your race, nationality, religion, political opinion, or membership in a particular social group. You can read more about asylum at http://myattorneyusa.com/asylum-and-refugee-protection.