In the standard bitcoin client (Bitcoin-Qt), they tell me the total money in the wallet. However, should I be able to see how much money is in each address?
Does bitcoin money belong to an address or on a wallet?
The bitcoins you received with an address are always associated with that address. The wallet is merely summing up the balances of individual addresses contained in the wallet. You can move addresses between wallets by exporting the private key from one wallet and importing them in another and the bitcoins will show up in the second wallet as well.