60-amp electrical service was common in homes built prior to 1950. Many century homes that were serviced with knob and tube wiring, converted to a 60-amp service.
Most homes today have a 100 or 200 amp service. But some still have only a 60 amp service. 60-amps is not enough to service a home with an electric stove, oven, clothes dryer, air conditioner, and so on.
Even if your home operates safely with a 60 amp service, the problem lies in insuring the home. If you are presently living in a home with a 60 amp service and you already have insurance, you may not need to upgrade. A home with a 60 amp service that has a gas stove and gas clothes dryer will have as much available power as a home with a 100 amp service that feeds major electrical appliances.
The concern has more to do with the branch circuit wiring. If you have very few outlets and find you have to snake extension cords throughout the house, an electrician may be able to add more circuits in the areas most needed.
Sometimes practicality forces a decision: if your electrical panel has no room for more circuits, upgrade the panel, or add a sub panel, and it may make more sense to upgrade the entire service at that point.
Insurance companies are concerned with the threat of overuse and overheating, increasing the risk of an electrical fire and a subsequent claim. Your insurance company may require you to upgrade your 60-amp service to 100-amps (the standard for new construction).
A good electrician can evaluate the entire system and recommend an upgrade path appropriate for your home and your needs.