Well, $100 minimum since I find it's usually enough to cover "emergencies" that might crop up before I can get to an ATM. Maximum, well, whatever I have with me. Normally only happens when someone re-imburses or pays me cash for something but I usually deposit that to my savings account sooner or later.
Not all places accept credit/debit cards (e.g. a lot of restaurants in Chinatown). Also experienced some cases when credit/debit card processing is down (their machine breaks down or is offline). It's just easier to always carry cash with me in those circumstances.
I never use my debit card (unless it's for ATM withdrawal).
Safer to use the credit card plus I get cash back for my purchases (just always pay in full every month to avoid interest). If someone from, say, a restaurant steals my credit card number, I'm not liable for the charges incurred. Someone steals my debit card number and it's possible they may drain my account and cause overdraft fees when checks I have issued bounce. It's more of a hassle getting your money back with debit cards. With credit cards, it wasn't your money to begin with.