Should you turn in a claim to your insurance company or pay it yourself?

It depends. If you're trying to keep your rates from going up, you'd need to look at your past claims history, the amount of damages from this loss, etc. If this is a minor loss, you can just pay out-of-pocket and be done with it. But, be warned: If you're talking about damages to another vehicle that you caused, you need to proceed very carefully. This is because the other party might take their car to a shop that finds all sorts of supplemental damages once the car is torn down. You would also owe the other party loss-of-use, or rental, which can get pretty pricey. You can always have your carrier handle the damages and reimburse them for any payments, but since you already pay for the insurance, I'm not sure why anyone would do this.