We all know that banks and stockbrokers can go bust. What happens to their customers in that case?

I would imagine that money in accounts may be lost, but shares/bonds in custody accounts would be returned to you (eventually).

Generally, the customers are insured up to some amount, for cash. For example, with interactive brokers, it is 100k $. If you have stock positions, you will be delivered the securities in paper form or they will be transferred to a broker of your liking.

If you have cash above 100k $, you might be in trouble for the rest of it.
Regards,
-Ip