No, you do not have to worry about new popcorn ceilings having asbestos. All residential building materials used since the mid-1980s stopped including asbestos, so new popcorn ceilings have no risk of asbestos. You can still create a textured drywall ceiling with a popcorn texture if you like, but new techniques have no risk of adding asbestos. 

There are products that still contain asbestos, but they were typically installed decades ago. Common places where you may still find asbestos-containing materials are in siding, wrapped around ductwork, in older appliances (i.e., stove and gravity furnaces), and in floors. Asbestos was primarily used as an insulator, so anywhere in an older home where you lose heat is where you will most likely find asbestos.

If you have an older electrical system with fraying wires and other signs of damage, removal  may be the best course of action. When asbestos begins to break down, it's more likely to become airborne, increasing the risk of health issues when it's inhaled.

However, if the system is in good condition and doesn't have visible signs of wear or damage, leaving it alone may be the safest course of action. An asbestos abatement and removal professional can help you determine the best way to proceed.

Unfortunately, asbestos is not easily identified and can often be hidden in plain sight in your home. While it can look like fluffy fibers, tiny pieces can also appear in materials like plastic or cement, making it impossible to identify on your own properly.

Plus, since asbestos is so dangerous when disturbed, you should never root around to try and find it. This job is best for a professional who can test for the material properly.