The only type of insurance that is always compulsory in relation to your mortgage is buildings insurance. However, if your property is leasehold then it is usually the freeholder that insures the property, so you may not need to arrange your own buildings cover.
The cost of the insurance, however, is effectively passed on to you as ground rents and service charges. If your property is freehold then it is your responsibility to arrange the cover: a point to note is that occasionally lenders will insist that you take your buildings insurance through themselves, or an insurer of their choice, but this is fairly rare.
Lenders also tend to offer other types of insurances along with the mortgage as an optional all-in-one package, although this is rarely the cheapest option. Mortgage payment protection for example ensures that if you are unable to work because of sickness, accident or unemployment your mortgage payments will be covered over the short term.
Permanent Health Insurance on the other hand provides longer term income protection but does not include redundancy cover. For many people, particularly those who have dependants, or those who are sole earners in families, life and critical illness insurance are essential when taking out a mortgage. Although attitudes to risk vary widely, it is always vital that you know exactly what insurances are available to you so that you can make an informed decision as to how much cover you need.
With all insurances it is always worth shopping around for the best deal, as premiums can vary hugely from insurer to insurer. As well as protecting your property it is also vitally important to protect your ability to repay the mortgage in the event of an unexpected event.