Our kids don't get a weekly allowance. You might think this is mean but I believe money needs to be earned in some way rather than just handed out.
First, we don't pay for chores around the house. That leads to the question "How much will I get?" whenever you ask them to do something. If they ask me that, I tell them how much they owe me for renting their room. The kids live in the house so they have an obligation to share in the domestic chores they're capable of. No arguments.
I will sometimes offer to pay them for a job that they aren't normally involved with such as washing the car or picking weeds. Its usually something outside to get them out of the house. Or I might pay them to do something like making greeting cards. If they can save me money, they can keep the money that's saved.
They don't get paid to practice their musical instruments. That's a commitment they made when they decided to learn them.
The kids go to "work" five days a week learning at school. I pay for results at school. At the moment, I pay for every sticker they receive when they've done a good job on an assignment or test. The tricky part is to come up with some quantitative and fair system that encourages them to put in extra effort.
I don't pay cash. We have an account book and I act like a bank. It's a generous bank that pays 1% interest on their accounts every month. It encourages them to save and shows them how money grows. Some day I'll teach them that interest rates change! I'm also an ATM machine.
If they want to buy something that I think is a good idea, I might offer to split the cost 50/50 with them. We negotiate all kinds of bargains and often require the kids to "sell" to us why we should share costs. The core idea here is that they must contribute some of their hard earned money. Clothes and books are an exception. We buy those. If we go to a fair or amusement park, I'll give them a budget and after they've spent that, they have to use their own money.
Sometimes they want to buy something that I would rather they didn't. I'll tell them why its a waste of money but after that, I usually bite my tongue. At that point, it's their money and they've earned the right to spend it as they wish.
This system has worked very well for us. The kids don't nag us for stuff and are careful with how they spend their money.