Students who are in their first year of education in the U.S. on the F1 Visa are not supposed to seek off-campus employment to support their living expenses in the country. For this, the applicants need to approach the U.S. Citizenship & Immigration Services (USCIS) which may grant the permission to work off-campus in very few circumstances.
Students can work on their respective campuses without seeking any exclusive permission from USCIS. A student can work for 20 hours a week during the course commencement and during semester breaks, the students can work up to 40 hours a week like any other full-time employee.
The exceptions which can be off-campus and may be most likely accepted by USCIS for the students to work are fellowships, assistantships, research programs etc.
Talking about off-campus employment, the designated school official comes into picture in approving permission for students to work off-campus. However, such approvals are solely at the discretion of the designated school officials. Even that is eligible for an application only when a student has crossed a minimum of 1 year of stay on the campus on the course.
Any kind of breaching of regulations directly result in deportation of students and such breach would be like illegal off-campus working like working in hotels, gas station, grocery stores etc.