How much will your insurance rate go up after a minor accident that was your fault?

The short answer is 'depends.' The long answer is that it depends on you age, how long you've been driving, previous driving history, and insurance company. If by minor you mean under $1000 property damage and under $750 bodily injury, then you may expect a 0-25% increase in premiums. If it's over those limits, you could see a much, much higher increase which could include policy cancellation. It really depends on your situation.