No minimum time off in the US. For most white collar jobs, 10 or 15 vacation days is typical (but many Americans feel pressure to not use all their vacation days). Some companies offer more, especially to employees with seniority. Government jobs tend to be nice about this, too. This is in addition to holidays -- about 8 or so per year is typical for these workers, I think (days like Christmas that most everyone in the company has off).
Those working blue collar jobs, however, aren't always so lucky. They might not have any vacation days or even holidays.
It's 28 days minimum in the UK[1] (for a full time employee) but these can include the public holidays (8 a year in England/Wales, 9 a year in Scotland and 10 a year in NI).
<generalisations follow>
In reality most IT companies give ~25 days and give the 8 public holidays off too. Longer service often gets you more days. I'm up to 27 days/year (plus the 8 public holidays).
Any reasonable IT company will also require you to take all of your holiday during the year, it's a manager's responsibility to ensure that his team are using their holiday. There's certainly no pressure to not use all of the days! I can carry over the odd day or two across years but generally not allowed to accrue large amounts of unused paid leave.
We don't have a concept of an allocation of "sick days" either, if you're unwell for a day or a few days you can self certify yourself off work sick. If you're ill for longer than a week the expectation is that you'll provide a letter from a Doctor explaining the absence. If you're ill frequently then the company will discuss it with you (and if you're pulling frequent sickies then that'll be dealt with as a disciplinary). I've had 3 days off sick in the last 5 years.
In Poland it's 20 or 26 days off, all state holidays are added on top of that (there are few days, probably around 8). All of those are paid. Except those one can also take unpaid vacation but that has to be negotiated with the employer.
20 days is until you have 10 years of work experience, but graduating counts as 8 years of experience so most people just have to work 2 years to jump to 26 days.
Those working blue collar jobs, however, aren't always so lucky. They might not have any vacation days or even holidays.