Too Many Claims
The average homeowner only files a claim around once a decade. That doesn't mean that filing twice in ten years will see you being dropped, but if you're filing on an annual basis, that's going to raise some red flags with your provider. Insurance is best reserved for severe damages that cannot easily be handled out of pocket.
Failing to Make Necessary Repairs
Your insurer may ask you to do some critical maintenance in order to remain covered. More times than not, this means the roof. An old roof can lead to cumulative water damage, and many insurers feel that someone who won't replace an old roof is too big of a risk to cover.
Your new pet is probably not going to have you looking for a new insurer, but some insurers will make exceptions for certain dog breeds and high-risk pets. And insurers will almost certainly drop you if multiple dog-bite liability claims have been filed. It's simply a legal risk that many insurers feel is not worth it.
Other reasons may simply be out of your hands. If your insurer goes out of business or decides not to cover your state anymore, you're going to need to go shopping for a new home insurance provider.
To avoid being dropped, your best course of action is to make sure that your home is in good shape, and to only rely on home insurance for disasters and major expenses. You don't need to file a claim every time a window breaks or a garage door is dented. Insurance isn't really a maintenance fund, it's a "just in case" safety net. So long as that's how you use your policy, you should have no problems staying covered.