Again, quoted passages are from Human Action Chapter 17 section 12.
Gresham's Law is often stated: bad money drives out good. This is incomplete, and an incomplete statement of Gresham's Law. It is more accurately stated: Bad money drives out good if their exchange rate is set by law.
If different "money" is free to trade, without any hindrance from legal tender laws, tax preferences, etc., bad money will not drive out good, nor necessarily will good money drive out bad. Different "monies" will find their value in the market, and will trade. Period.
However, if one money is protected by legal tender, and the valuation is fixed by government, then yes, the bad money will drive out the good. All will get rid of the over-valued (by government decree) money, as it MUST be accepted by the counter-party, and accepted at a value higher than the market would determine.
Mises addresses this idea as follows:
"If, however, the things concerned are not money-substitutes and are traded at a discount below their face value, the assignment of legal tender quality is tantamount to an authoritarian price ceiling, the fixing of a maximum price for gold and foreign exchange and of a minimum price for the things which are no longer money-substitutes but either credit money or fiat money. Then the effects appear which Gresham's Law describes."
If, by decree, the "thing" that is no longer a money substitute must be accepted not at the discounted value of the market, but at the inflated value of the decree. This money will always be spent first, and it will drive the truly "good" money out of circulation. Why would someone spend a 100 cent dollar when a 90 cent dollar must be accepted by the counter-party at 100 cents? Bad money will drive out good money.
Mises then addresses what would happen in a free banking environment, where competing currencies trade freely without legal tender. In fact, he is for the more the merrier, as in the following:
"However, freedom in the issuance of banknotes would have narrowed down the use of banknotes considerably if it had not entirely suppressed it. It was this idea which Cernuschi advanced in the hearings of the French Banking Inquiry on October 24, 1865: "I believe that what is called freedom of banking would result in a total suppression of banknotes in France. I want to give everybody the right to issue banknotes so that nobody should take any banknotes any longer."
He proposes that the good money will drive out the bad. In a freely competitive environment, where multiple banknotes would be issued, he sees the entire concept of banknotes disappearing, in favor, presumably, of good coin.
I think this will be true for the absolutely worst banknotes. However, it strikes me that most will continue to trade, but not at face value. Perhaps this is what Mises is getting at with the following:
"But if any doubts exist concerning their prime character, people will hurry to get rid of them as soon as possible. They will keep in their cash holdings money and such money-substitutes as they consider perfectly safe and will dispose of the suspect banknotes."
For safety, people will keep the most secure notes, the notes issued by banks with the best reputation. They will preferably spend the suspect notes, or more likely take them to the issuing bank for 100 per cent redemption.
Such will be the check on any bank, and this I return full circle to Mises and free-banking. He sees the market perfectly capable of "policing" if you will, the various banks participating in the market. Once the reputation of a bank is in question, notes will be brought to the bank to be redeemed. Unless the bank is acting properly in its reserves, it will soon find itself depleted. The end comes quickly when the reputation is in question.