Not often allowed.
Routinely allowed in my experience.
Gotta be careful about whether we’re talking about 1970s practice or 2020s practice.
I was talking about 1970s and 80s. It was a normal regular thing.
But presumably it was more along the lines of - people paid in twenties, then the business took them to the bank. Meaning, if the bank did actually (manually?) check for serial numbers, they would have a large pile to go through. My memory of finances was in 1968 was being astounded that my step-sister ran up $100 for weekly groceries for 6 people in Long Island. And how many cheques here there and everywhere my brother-in-law wrote then, in the days before ubiquitous credit cards.
It was also a normal thing to use your supermarket as a bank.
You could buy $42 of groceries, write a check for $62 , and the cashier gave you $20 in cash.
And some places would let you cash your entire paycheck.You handed them your paycheck from work, endorsed it on the back " pay to the order of Kroger, inc", and the cashier gave you your week’s salary in cash.
You can still do that with a bank card. At one grocery I go to, the PoS device asks if you want change back. That grocery only accepts debit cards, not credit (the fees are lower on debit cards), but they will give back cash. One annoyance, though. They have their system set up so that the cashier has to confirm you want or don’t want cash back twice during each transaction (once before and once after you’ve entered your PIN). Not sure why, but if the cashier isn’t paying attention, you have to wait a while until they do.
This is common in Germany, with debit cards in lieu of checks. It’s a society where a lot of people still use cash to pay for their groceries, so this is a way for the supermarket to reduce its intraday cash holdings.
That was certainly a thing here in Oz a decade or so ago. Supermarkets would always offer additional cash out when doing a purchase with a card. Did need to be a debit card. It was, similarly, a good way to keep cash levels down. I can’t remember when I last saw it explicitly offered, but it may well still exist for anyone needing cash. Cash here is fast vanishing. I’m not sure I remember the last time I used any.