Applying to new grad RN program but currently employed at a snf

Yes. You are a new grad and had to take what was offered. They all know this. Your current employer doesn't know when you applied for the new opportunity and probably doesn't even care....they needed a slot filled and obviously nurses were not lining up at the door for it if a brand new nurse was hired into a supervisory role. Do what you have to do. You landed that position because the last nurse vacated it. Your career will be molded by the things you learn....or don't. The latter will limit you. 

I would leave it off the resume. Three weeks does not count as experience, and could only open up questions like, why are you leaving so quickly? 

 You are 5 months post licensure. If they asked why  you haven't worked yet, tell them you are looking for a good fit, and you are confident this it.

Been there,done that said:

I would leave it off the resume. Three weeks does not count as experience, and could only open up questions like, why are you leaving so quickly? 

 You are 5 months post licensure. If they asked why  you haven't worked yet, tell them you are looking for a good fit, and you are confident this it.

Leaving it off when the application specifically states to list all employment would constitute falsification of a job application.  Certainly saying you "haven't worked yet" would as well.   Both these actions are considered grounds for immediate dismissal.  If discovered during the hiring process it would result in you not getting the job, if discovered after employment it could result in you being fired.  And before you are tempted to think they'll never know, don't forget they do a background check which often lists previous employers.  
 

A safe answer would be that you preferred inpatient but only took that job since the inpatient position fell through.  Then circle back to wanting to learn more advanced skills in the inpatient setting (or something along those lines).  Personally I don't think a new grad should have been made a RN supervisor (although to my understanding of LTC that seems to mean "only RN in the building" more than an actual supervisory role).  Regardless I'd avoid mentioning it or do so carefully as you don't want to come across as criticizing your previous employer's hiring and staffing (even if such criticism is well founded!). As a rule, employers are hesitant to hire potential employees who criticize their former employers.  Using the approach that you would prefer inpatient avoids this.  I'd also use the same approach when asking your current manager for a reference (which you should do before listing them!).  Good luck and best wishes for a great new job!  Keep us posted! 

Update: Thank you for your replies! I have two  interviews with two different hospitals for new grad programs, so I'm hoping for the best. As for the SNF job, I had to put in my 2 weeks notice because I felt I wasn't getting enough support/training (trained with an RN for one day) and resources to keep practicing there as a new nurse.  I want to build a strong foundation and practice safely, but unfortunately that wasn't something I could receive there.