No broker will last forever, but fortunately, that is not too important.
This applies to stocks and ETF's bought on US exchanges through a US broker, so I believe it will apply to you. But it is easy to check.
Simply, most stocks and ETF's bought in the US are not held by the broker, nor by the customer. They are held at Depository Trust Company (DTC), which was established precisely to manage the situation of where the asset will be.
Where you stand in case a broker fails can get complicated.
However, the case of MF Global is totally different. MF Global was a futures broker. A futures contract is not an actual asset -- you don't own anything. You have margin with your broker to guarantee that you will have funds for your account to be marked to market at the end of the day. Cash goes in and out of your account based on price fluctuations, but you don't actually own, let's say, a piece of the S&P 500, or any gold or wheat. In the case of physical commodities like gold or oil, you can take physical delivery on settlement (and pay the purchase price, which is much more than the margin), but until then you don't actually own any, and most traders never do. So all you've got with the broker is the cash margin in your account. You'll have a legal claim if the broker fails, but that doesn't always work out well.
An equity broker is doing something entirely different. When you buy a stock or an ETF, you own something: x number of actual shares. Interactive Brokers is both a futures and an equities broker (and forex). But their businesses will be separated and will come under different financial regulators. And when you buy a stock or ETF, it's an asset that has an actual existence.
Getting it if there's a problem may be another thing.
If you buy a stock or ETF, it usually will be held in "street name" at DTC. This can get complex, but it means that DTC doesn't have you down as the owner. (See here for more: https://en.wikipedia.org/wiki/Street_name_securities ) So it depends on broker account records.
Here are some things to be aware of:
1. Theoretically, there may be no problem connecting the DTC holding to you, but it would be a good idea not to have the assets held in a margin account with your broker. Why? "Margin" is very different in the equity/ETF world compared to the futures world. All futures accounts have "margin", in fact, that's all they have . But "margin" is not the same for equities. To buy a security on margin means that you put up the required margin amount (currently 50% of the purchase price), and the broker lends you the rest. When you open a margin account in this sense, you will also sign an agreement that lets the broker use the shares for other purposes, such as to loan out the shares to another party for short sales. Not to get too far into the topic, but if you're interested in safety, don't do that. You want to be sure they will be there.