I was in the United States Navy and I want to know if I can enlist in the Singapore military?
You can as a permanent resident but most likely the employer who hires you to work in Singapore would make you complete your work contract before you enlist.
If permanent residency is required at minimal, then how do I get permanent residency?
You need to get employed by a Singapore company or individual that would sponsor your permanent residency.
Also, I've had pain in my knees for the past 2 years, but haven't done surgery or anything to it. Would that disqualify me from doing Singapore National Service?
They will review you and see what you are capable of doing for BMT. If you cannot run for 2.4 km (6 track laps) then they will put you in a shortened BMT but you still become a soldier.
Before you start buying a plane ticket and move there, consider that Singapore is more expensive then New York City. Alot of Singapore NSmen get financial aid from their families and leave after 2 years due to the fact that they do not get a salary that can pay the bills. Also consider that Singapore is full of Asians. If you are not Asian then it would be a huge culture shock. Living in Singapore does take time to accumulate as it is hot and humid 24/7. Just imagine living in a sauna. I'm not sure why you exactly want to serve in the Singapore military but just take note that Singapore is very strict in that you WILL be arrested for chewing gum. It is in fact a very boring city in that there are few places to have fun other than the countless shopping malls. Consider the job you will be working in order to get a permanent residency. If you get a low income job you will be working with foreign migrants that know little English. Trying to quit that job is very hard as your employers can revoke your permanent residency and thus you cannot serve. Search for a job BEFORE you leave for Singapore and search for one that matched the skills you had in the Navy. Singapore really needs skilled workers and they actually treat you better as a skilled worker. You'll probably have to serve a 2 year contract working for that company though but after that you can try to serve in NS.
My dad and grandfather were Singapore PRs.