I've heard horror stories about credit card insurance. A quick Google search discovered people complaining that they'd paid monthly fees to their credit card company, with the understanding that this would purchase them a break in their monthly payments if they ever became sick or unemployed. To hear them tell it, credit card companies loved this arrangement - but only as long as they were the ones receiving the money. When the cardholder actually did become sick or unemployed - the credit card company suddenly decided they didn't want to honor the agreement!
For that reason, I think it's misleading to even use the word "insurance" for the products that are being offered. While the actual insurance industry is one of the most heavily regulated industries - because they're covering major purchases like your home or your health - there's not a similar set of guidelines covering credit card "insurance". Instead banks have apparently adopted the word insurance to make you think that you're getting an equal level of protection for your credit payments. But you're not. You're getting an exotic new arrangement that your individual bank dreamed up for themselves, and decided to call "insurance."
Having said that, the types of "insurance" they'll offer you fall into a few major categories.
* Purchase protection plans offer to reimburse you if you've purchased an expensive item which later becomes damaged. Just remember that this protection plan isn't free, so you'll want to add up how much you'll be paying your bank each year for it first. You might discover that the cost of the plan represents a big percentage of whatever amount you could ever hope to collect in the event that your purchase was damaged. And when you try to collect that amount of money, you'll want to be sure that you understand exactly what your plan is covering - just in case your bank suddenly becomes reluctant to actually honor their policy!
* Unemployment insurance. Some credit card companies offer to suspend your payments in the event that you're unemployed. For some people, this sounds like a better deal than it is. If you've run up a huge credit card balance, you're still going to have to pay it off. Getting a temporary reprieve while you're unemployed doesn't change the fact that you're eventually going to have to find a job and pay off the balance. And if you aren't able to find a job - well, that suspension of payments isn't permanent!
* Disability insurance is basically the same as unemployment insurance: the credit card issue offers a break in the payments in the event that you're disabled and thus unable to make payments. The big problem with this is that it's the bank who defines "disabled," not a government agency. Their definition and yours might not agree!
* Identity theft insurance. Remember that it's already illegal to steal someone's identity, and banks are required by law to honor your disputes of any purchases that they've mistakenly billed to your account. So yes, there's horror stories about how long it is to recover your identity - but it's not clear how "identity theft" insurance really helps the situation. Most of the services that they offer you can be purchased relatively cheaply in the event of identity theft - like credit report monitoring - and think about what that really says about the program. It's not like those services make you any less vulnerable to identity theft in the first place!