In the US, I had two jobs that had vacation, sick and holiday time completely differently.
Job 1: 25 paid days that began to accrue from the day you began work; these days were for vacation or for sick time. If you were sick for 25 days, you didn't get any additional time unless you were able to apply for a long-term leave of absence (generally: chemotherapy, chronic or terminal illness). We got about 8 US paid holidays, but most of them we were required to work anyway.
Job 2: Unlimited sick time (although if you got out of hand, your supervisors would monitor). Everyone automatically got 10 vacation days per year, which do not roll forward; for every 5 years of experience after that, you would get an additional 5 days. So, when I had 6 years of work experience I got 15 vacation days. This company also gave us 10 paid holidays and would randomly e-mail us that we wouldn't have to come in on certain days.
In summary: in the US, with the exception of FMLA and disability leave which is federally regulated, personal time, sick time, and holidays are completely discretionary and vary by company.