I won't go through them all, but they faced the same problems we face today, and the same actors are the cause:
In the panic of 1873, lots of railroads had been financing up to the hilt with debt because you couldn't lose money building railroads (or houses, right?) and they lobbied the government for easy money policies (non-convertable notes, issuing more currency) so that the value of the greenback would drop, making their outstanding debt worth less in real terms.
Then the supreme court ruled in Hepburn v. Griswold in 1870 that "Legal Tender" (non-convertable US notes) was unconstitutional as per article 1 by a 5-3 vote. That shook up the market for the greenback, which was already trading at a heavy discount to gold dollars. But, don't worry, sure to create fear in the markets, President Grant appointed two new Republican judges (one open spot, and another spot created by a Majority member retiring post-vote), who quickly overturned that decision in Knox v. Lee by a 5-4 vote in May 1871.
Add on top of this divide between wanting Gold backed dollars vs US non-backed "greenbacks", the money supply doubled as new banks opened and loaned out too much. According to Murray Rothbard:
"The panic of 1873 was, not to initiate a great depression, but to cause bankrupt- cies in overinflated banks and in railroads riding on the tide of vast government subsidy and bank speculation."
The trigger seems less important than the fundamental problem of bank runs, though, which is as old as banking: some trigger will eventually come by, and then you're hosed. The only way besides deposit-insurance I can think of to avoid it is full-reserve banking, where a bank literally holds its depositors' money in the vaults. But such banking has never been commercially very successful, because it has to charge fees and pays no interest to depositors, and when there hasn't been a crisis in some years, depositors' memories grow short, only to be awaked again in panic when it seems another crisis might loom.
One possibility could be to mandate that normal retail banking be full-reserve, by prohibiting banks that take retail deposits from lending that money out. That'd be fairly draconian regulation, though, even more strict than the old Glass-Steagall regulation of what retail banks could do with their deposits. (It seems Rothbard wants to get rid of fractional-reserve banking without the government banning it, but I'm skeptical it would actually go away the way he thinks it would.)
As far as full reserve banking goes, I think that it is all too easy to forget what bank products were actually intended for and how they are used now. For instance, a bank could in theory offer full-reserve banking without charging a fee to its customers by utilizing age-old banking products: The fixed term Certificate of Deposit (CD) and the Savings account.
If you wanted to invest your savings into something that would draw interest, you would use a CD, and the bank would be able to lend out that money for the designated length of maturity of the CD. Yea, mortgage interest rates wouldn't be 4%, but the proliferation of Buy now, pay later attitudes would decrease also.
If you didn't want to tie up your money for a fixed length of time, you could put it into a savings account. No way, other than by transferring to another account could you get money from this account. The bank could use this for shorter-term liquid investments (treasuries, whatever).
Checking accounts and accounts with access to credit / debit cards would require a fee and the bank could not lend out those funds, as they are held in full-reserve.
Ta-da, full reserve banking (not sure how Rothbard would have felt about the savings account thing). Way off-topic from the OP, but a great debate non-the-less.
I find this discussion very relevant to the question of whether Paypal is a bank, whether it should be a bank, if we were better off if Paypal was brought under government banking regulations, etc.
They were never very good at that business, though. The main problem is that even a very conservative bank can't survive a large-scale bank run, and bank runs happened about every 15-20 years for an extended period of time: in the U.S., the Panic of 1819, Panic of 1837, Panic of 1857, Panic of 1873, Panic of 1893, Panic of 1907, and finally the panics of the late 1920s.