A good place to start is the home's foundations. The home should be securely fixed to its foundations. It's also important to check for signs of deterioration such as crumbling or cracking in cement foundations. Any ground movement could make this problem a lot worse.
Chimneys are another area that can become easily damaged during an earth tremor. This is very dangerous as the chimney may run the risk of falling in on itself, causing serious harm to anyone in the vicinity. If you have an unreinforced masonry chimney, it would be a good idea to secure it, which you can do using sheet metal straps and angle bracing. It would also be wise to avoid having children’s playrooms or play areas near the chimney.
Do you have many large, freestanding appliances in the home, like a water heater, for example? If so, you will want to fix these to a wall which you can do using metal straps.
Gas leaks are a common and serious problem after an earthquake. Again, fix any appliances that run on natural gas to the wall or floor. Unless you have an automatic shut off gas valve, it's very important that you clearly label the manual shutoff valve and always have easy access to it so that, if you need to, it can be shut off quickly.
These are just some of the areas in the home that may need securing so that they do not become a danger during a minor earthquake. You may also benefit from a home inspection which can help you make a proper assessment.