What free banking might look like http://www.learnliberty.org/videos/what-free-banking-and-why...
TLDR: Banks regularly collapsed and not just because of fraud, but started to collapse less as they got regulated more. The system functioned (for certain values of "functioned"), but shockingly enough the man on the street did not turn out to be better at evaluating a bank's solvency than central banks. They were of course also less able to bail themselves out if they misjudged a bank's solvency.
I don't really see why fractional reserve banking is necessary.
Also, since most full reserve proposals don't ban credit creation altogether (i.e. you can still issue and exchange IOU notes), but simply restrict credit creation and acceptance by institutions calling themselves banks, you end up with the economy relying on an unregulated shadow banking system even more prone to booms and busts to fulfil credit needs.
Theoretically if it is that much of an anathema one could have a 1:1 reserve and explicit money market style transactions for everything but the question is what would be gained? The liquid cash would be "safe" from bank runs at the cost of guaranteed losses to depreciation and storage fees. Keeping it insured and interest generating is a winner as an option.
1. Depositors will have to explicitly give up/lock their funds for a set period of time if the funds are to be lent out and they are to earn interest. Early withdrawals come with a significant fee. This makes bank runs very unlikely, eliminating the need for a central bank. The longer you want to lend the money for, the higher the interest. But you can also lend money overnight.
2. No money creation, leading to proper alignment of time preferences of investors/consumers and businesses. Interest rates correctly reflect this time preference. If money represents a claim on real resources, then a business can only use as many of these resources for investment as consumers are willing to forego consuming. Interest rates reflect the actual supply and demand of resources/consumption at different time horizons and do not get artificially "set" or otherwise manipulated.
Because of these two properties, the boom-bust leverage cycle is greatly dampened, if not completely eliminated.
(1) Because maturity transformation is no longer permitted, every time somebody wishes to take out, say, a mortgage, the bank has to find somebody who has >$200k they're not even going to consider spending for quarter of a century. As a result, costs of borrowing go massively up, which is bad for everybody except for a small proportion of rentiers sufficiently wealthy to be entirely unworried by liquidity. Joe Sixpack proves to be even less adept at managing an overnight lending portfolio than full time professional bankers, and loses money as a result.
(2) The money supply is artificially "set" at a particular level on the basis of the unambiguously false assumption that real resources do not change over time rather than being allowed to fluctuate and grow to properly align itself with creditworthy borrowers' growth projections in a way commensurate with price stability. The boom bust cycle is replaced by permanent bust.
2. The money supply could grow predictably, e.g. like Friedman's proposal of replacing the Fed with a computer that expands the money supply in a predictable manner.
2. Friedman's k% rule is better than a fixed money supply, but it's still every bit as arbitrary and further removed from the relevant market indicators of resource constraints (creditworthy borrower demand and price inflation) than the current system.