Pet insurance is very common here in the UK, and without it - many dogs would not be getting spinal surgery, MRIs, myelographs, TPLOs, weeks of ICU hospital care, etc. at our vet school (or anywhere else). There are several types of insurance here. Some set a limit of money provided for the lifetime of the animal. Others pay out a certain amount each year. Then there's the type of insurance you can buy for valuable breeding or performance animals (i.e. race horses) that will supply a certain amount of money if the animal is injured or becomes infertile/incapable of breeding.
I have heard that in the US, the insurance companies tend to try to avoid paying for pets' problems by "blaming it on the breed". If a breed is predisposed to hip dysplasia, for instance, then they will refuse to pay for total hip replacement (for example - not sure if this is real or not). There are apparently a huge list of disorders that insurance companies refuse to cover due to the breed of the animal. I'm not sure how it would work for a mixed breed.
I don't have much of an opinion at the moment in the US because I still haven't found that many pets are insured, and I think it will be a long time before that happens. In the UK, I see pet insurance as a great thing.