When I worked public safety
We started the year with 3 personal days, could carry over 3 days if I remember right. So 6 days off the bat.
Holidays if you worked could be saved as a day off, (which I did) or taken as time and a half. A day off at straight time was worth more to me than time and 1/2.
Vacation was earned, and at increasing amounts for length of service.
Sick time was in a separate "account" from all the above.
And, I worked a set schedule, every Monday and Friday 24 hours. So most holidays fell on these days. I'd take a day and be off a week.
We could also "sell" unused personal days back to the county.
Give you an idea of how much time I have/had. I retired close to 7-8 years ago. I went back into the job part time still in the state retirement system, so I kept my time. I have roughly 900 hours sick time and like a 150 hours vacation time.
Its confusing but, this is as close to unlimited time as I've heard of.